Bloch-Kato Conjecture for the Adjoint of H¹(X₀(N)) with Integral Hecke Algebra Citation Lin, Qiang (2004) Bloch-Kato Conjecture for the Adjoint of H¹(X₀(N)) with Integral Hecke Algebra. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/QD4X-J291. https://resolver.caltech.edu/CaltechETD:etd-11182003-084742 Abstract Let M be a motive that is defined over a number field and admits an action of a finite dimensional semisimple Q-algebra T. David Burns and Matthias Flach formulated a conjecture, which depends on a choice of Z-order T in T, for the leading coefficient of the Taylor expansion at 0 of the T-equivariant L-function of M. For primes l outside a finite set we prove the l-primary part of this conjecture for the specific case where M is the trace zero part of the adjoint of H¹(X₀(N)) for prime N and where T is the (commutative) integral Hecke algebra for cusp forms of weight 2 and the congruence group Γ₀(N), thus providing one of the first nontrivial supporting examples for the conjecture in a geometric situation where T is not the maximal order of T. We also compare two Selmer groups, one of which appears in Bloch-Kato conjecture and the other a slight variant of what is defined by A. Wiles. A result on the Fontaine-Laffaille modules with coefficients in a local ring finite free over Z ℓ is obtained. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: adjoint motives; Bloch-Kato conjecture; Burns-Flach conjecture; Modular forms Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Scott Russell Johnson Graduate Dissertation Prize in Mathematics, 2003. Thesis Availability: Public (worldwide access) Research Advisor(s): Flach, Matthias Thesis Committee: Flach, Matthias (chair) Ramakrishnan, Dinakar Goins, Edray Aschbacher, Michael Defense Date: 19 September 2003 Record Number: CaltechETD:etd-11182003-084742 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-11182003-084742 DOI: 10.7907/QD4X-J291 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 4595 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 06 Feb 2004 Last Modified: 20 Jan 2021 23:38 Thesis Files Preview PDF - Final Version See Usage Policy. 417kB Repository Staff Only: item control page

Bloch-Kato Conjecture For The Adjoint Of H 1(X0(N )) With Integral Hecke Algebra Thesis by Qiang Lin In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2004 (Defended September 19, 2003) ii c 2004 Qiang Lin All Rights Reserved iii Acknowledgements I would like to express to my advisor, Matthias Flach, my great appreciation for his suggestion of the topics, and for his continuous help during my graduate study and the preparation of this thesis. I would like to thank math department of Caltech and Caltech at large for providing a friendly, challenging and encouraging environment. iv Abstract Let M be a motive that is defined over a number field and admits an action of a finite dimensional semisimple Q-algebra T . David Burns and Matthias Flach formulated in [B-F3] a conjecture, which depends on a choice of Z-order T in T , for the leading coefficient of the Taylor expansion at 0 of the T -equivariant L-function of M . For primes ` outside a finite set we prove the `-primary part of this conjecture for the specific case where M is the trace zero part of the adjoint of H 1 (X0 (N )) for prime N and where T is the (commutative) integral Hecke algebra for cusp forms of weight 2 and the congruence group Γ0 (N ), thus providing one of the first nontrivial supporting examples for the conjecture in a geometric situation where T is not the maximal order of T . We also compare two Selmer groups, one of which appears in Bloch-Kato conjecture and the other a slight variant of what is defined by A. Wiles. A result on the Fontaine-Laffaille modules with coefficients in a local ring finite free over Z` is obtained. 2000 Mathematics Subject Classifications: Primary 11F67, 11F80, 11G40; Secondary 14G10, 19F27 Key word and phrases: Burns-Flach conjecture, modular forms, adjoint motives. v Contents Acknowledgements iii Abstract iv 1 Introduction to Burns-Flach conjecture 1 1.1 From history to this article . . . . . . . . . . . . . . . . . . . . 1 1.2 Motives with action of Hecke algebra . . . . . . . . . . . . . . 3 1.3 Explicit Deligne’s conjecture . . . . . . . . . . . . . . . . . . . 6 1.4 The Burns-Flach conjecture in our case . . . . . . . . . . . . . 10 2 An isomorphism between tangent space and Selmer group 15 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . 19 3 Reduction and proof of ad0 (H 1 (N ))(1) 29 3.1 A reformulation in classical terms . . . . . . . . . . . . . . . . 29 3.2 Reduction to a concrete problem . . . . . . . . . . . . . . . . 36 3.3 Differ by a unit in T` . . . . . . . . . . . . . . . . . . . . . . . 42 4 A non-trivial example 50 vi 5 An isomorphism of two Selmer groups 56 5.1 The theory of Fontaine and Laffaille . . . . . . . . . . . . . . . 57 5.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.3 The isomorphism of two Selmer groups . . . . . . . . . . . . . 68 Bibliography 80 vii List of Tables 4.1 Hecke operator Tn on newforms in S2 (Γ0 (89)) . . . . . . . . . . 53 1 Chapter 1 Introduction to Burns-Flach conjecture In this chapter we introduce explicitly Burns-Flach conjecture given in [B-F3, §4.3] (which uses the language of perfect complexes and their determinants to virtual objects) for the interested motive here. 1.1 From history to this article For simplicity, a motive always means the collective data of its various standard realizations and standard compatibility isomorphisms among them, which are essential in order to define its L-function and to state certain conjectures. It also goes by the name “motivic structure” in [F-P2] or “(S-integral) premotivic structure” in [D-F-G2, §1]. The special values of L-functions of motives at integers have long been an inspiring source of number theory. A recurring phenomenon is that the values of L-functions at integers reflect arithmetic properties of the objects used to define the motives. Two prominent examples are Dirichlet’s class number formula and Birch and Swinnerton-Dyer 2 conjecture on elliptic curves. There are vast amount of work, to compute and various conjectures to predict the value of a general L-function at zero for motive over Q. Conjectures of Deligne [De1] and of Beilinson [Be] achieve that up to a rational factor. Then the seminal conjecture of Bloch-Kato in [B-K] describes the precise value up to sign using cohomological data. Its refinements and reformulations by Fontaine, Kato and Perrin-Riou [Fo1, Ka2, F-P2] are further refined and generalized to a (non-commutative) equivariant version, namely, the equivariant Tamagawa number conjecture of Burns-Flach [B-F3, Conjecture 4], which shall be called “Burns-Flach conjecture” for brevity. For more history of, evidence for and relations among these conjectures and the Iwasawa main conjecture, see [B-F3], [B-F4] and [Hu-Ki]. While the title of this thesis is still paying tribute to Bloch and Kato, its emphasis is on the phrase “with the integral Hecke algebra,” indicating that we are actually dealing with Burns-Flach conjecture. A complete title should read as “proof of Bloch-Kato conjecture as refined by Burns-Flach except at finitely many places in the case of the trace zero part of the adjoint motive of H 1 (X0 (N )) for prime N with the action of integral Hecke algebra for cusp forms of weight 2 and the congruence group Γ0 (N ).” The upshot here is that the integral Hecke algebra is strictly contained in the maximal order of its maximal quotient, thus our proof can be viewed as one of the first examples where Burns-Flach conjecture is finer than previous conjectures. It is expected 3 that Burns-Flach conjecture, will remain open in general for years to come to challenge mathematicians. We assume that readers are familiar with the standard symbols in number theory and cohomology theory such as Z, Q, R, C, Fp , Zp , GK for a field K, H i (·, ·). In particular, H i (K, V ) for a field K and a continuous GK -module V is the i-th continuous cohomology of GK with coefficient in V . OR is the maximal order of a commutative algebra R over a number field. We use the notation of [D-F-G2, §1] for the various realizations of these motives, as well as for their integral versions by respective Gothic letters. For example, a motive M over a number field K contains M∗ for ∗ = B, dR and λ (a place of K), which denotes its Betti, de Rham, λ-adic and `-crystalline realization respectively, and correspondingly, its integral version M contains M∗ . We often use “=” to denote the canonical or understood identification between two objects. 1.2 Motives with action of Hecke algebra Let M be a motive over a number field with an action of a finite dimensional semisimple Q-algebra T . Let T ⊂ T be a Z-order such that there is a projective T-lattice in the realizations of M . The meaning of “projective” is given in [B-F3, Defintion 1 of §3.3]. Conjecture 4 of loc. cit. concerns the leading coefficient at s = 0 of the T -equivariant L-function L(M, s) of M . This is the aforementioned Burns-Flach conjecture, which depends not only on M and T but also on the choice of T. Roughly speaking, Burns-Flach conjecture spe- 4 cializes to Bloch-Kato conjecture when T is commutative with maximal order T. Let f (τ ) = P an e2πinτ be a (normalized) newform of congruence group Γ0 (N ), weight k ≥ 2. The number field Kf /Q is generated by the Fourier coefficients ai of f . We can attach a motive Mf over Kf and its integral version Mf to f as given in [D-F-G2, §5.4], thanks for the most part to Eichler, Shimura, Deligne, Jannsen, Scholl and Faltings. We reiterate here that a motive is treated naively as the collective data of its realizations and various compatibility isomorphisms among them. We take the trace zero endomorphisms of Mf to obtain the self-dual motive Af = ad0 Mf = HomKf (Mf , Mf )0 . (The upper index 0 denotes endomorphisms of trace 0.) In [D-F-G2, Theorem 8.10] the `-primary part of Bloch-Kato conjecture for Af , or equivalently, of Burns-Flach conjecture for M = Af , T = Kf and T = OKf , was proven with ` outside a certain finite set Sf of primes. See §1.4 for the definition of Sf . Instead of one newform, let us consider a finite set I of newforms f and put MI = M Mf f ∈I T = Y Kf f ∈I AI = HomT (MI , MI )0 = M f ∈I BI = AI (1) Af 5 This will be our basic setting, upon which much more symbols will be introduced along the way. Note that the above definitions of Af , MI , AI and BI are kind of symbolic definitions that abstract the natural construction of various components of a motive and also its integral version. MI , AI and BI are motives with the natural action of the semisimple Qalgebra T as Kf acts on Mf canonically for all f ∈ I. If we take direct product over I, the result of [D-F-G2, Theorem 8.10] also gives the Burns-Flach conjecture for M = AI , BI and T = OT = Q f ∈I OKf . However, in many natural cases we have the following finer structure. There is an appropriate integral Hecke algebra T that is generated by Hecke operators over Z, or equivalently, by Hecke correspondences over Z. The motive MI carries a lattice, also called the integral version of MI , which is projective over T. We have a canonical map T → T : t 7→ (a1 (t(f ))f ∈I , where a1 (g) is the first Fourier coefficient of the q-expansion of a newform g and which is consistent with their action on the motive MI , AI and BI . Hence T may be (considered as) contained in, but generally different from, the maximal order of T . This difference will be seen to be a key fact that the importance of this article relies on. A typical example, which we actually work on in Chapter 3 and Chapter 4, is where I is a complete set of orbit representatives for the action of GQ on newforms of weight 2 and congruence group Γ0 (N ) with prime N . Then MI = H 1 (X0 (N )), and the integral Hecke algebra T generated by Hecke correspondences of X0 (N ) over Z is well known to be an order in T , which can 6 be identified with the set of all correspondences over Q, or with the rational Hecke algebra for cusp forms of weight 2 and the congruence group Γ0 (N ). Mazur proved in [Ma1, II, (14.2), (16.3), (15.1)] that the integral cohomology H 1 (X0 (N ), Z) of X0 (N ) is locally free (i.e., projective) over T except possibly at maximal ideals containing 2. (Just for the record, this exception at the place over 2 might be necessary as indicated by [Ki].) In this case T also coincides with the full endomorphism ring of the Jacobian J0 (N ) of X0 (N ), which is proven in [Ri, Corollary 3.3], and so it is the natural ring to consider when studying the motive H 1 (X0 (N )) = H 1 (J0 (N )). We will formulate Burns-Flach conjecture for BI with the action of T with the choice of projective T[1/2]-structure B that comes from the projective T[1/2]-module H 1 (X0 (N )(C), Z[1/2]), or more explicitly, from the integral premotivic structure of level N and trivial character in [D-F-G2, $4.5]. We shall focus on BI rather than AI because the methods there apply more directly to BI (see in particular [D-F-G2, Lemma 8.11]). Once we proved the case on BI , the desired result on AI can be formulated similarly and can be seen along the same line of arguments. 1.3 Explicit Deligne’s conjecture To prepare for the statement of the Burns-Flach conjecture in $1.4, we introduce in this section Theorem 8.5 of [D-F-G2] that is an explicit form of Deligne’s conjecture. 7 With the index set I implicitly understood, we will write M, A, B, T instead of MI , AI , BI , TI . Throughout this article the L-functions of A and B, L(A, s) and L(B, s), are their T -equivariant L-functions, whose definition is given in [D-F-G2, §4] and in particular, its Remark 7. Recall that IKf was defined in [D-F-G2, §1.2] as the set of embeddings Hom(Kf , C). We let IT = Hom(T, C) denote the set of ring homomorphisms T → C and we then have IT = ` f ∈I IKf . We view L(Af , s) as the tuple of functions L(Af , τ, s)τ ∈IK , i.e., as a holomorphic function with values in (Kf )C := Kf ⊗Q C = CIKf . Similarly we view L(A, s) as a function with values in TC := T ⊗Q C ∼ = CIT . The special values L(A, 0) and L(B, 0) = L(A, 1) then lie in TR := T ⊗Q R. We revert M temporarily to mean a motive with coefficients in K so as to introduce several symbols the last of which is Deligne’s period c+ (M ). The fundamental line for M is the K-line defined by ∆(M ) = homK (det MB+ , det tM ) K K where + indicates the subspace fixed by F∞ , the action of complex conjugation, and tM = MdR / Fil0 MdR . Furthermore the composition of (I ∞ )−1 R ⊗ MB+ → (C ⊗ MB )+ −→ R ⊗ MdR → R ⊗ tM 8 is an R ⊗ K-linear isomorphism. Its determinant over R ⊗ K, called Deligne’s period of M and denoted by c+ (M ), defines a basis for R ⊗ ∆(M ). Now we quote verbatim Theorem 8.5 of [D-F-G2], which is an explicit form of Deligne’s conjecture in the cases of Af and Bf for a newform f of weight k, conductor Nf , character ψ, and field of definition Kf . Theorem 1.3.1. Let b(Af ) ∈ ∆(Af ) be defined by the formula hf, b(Af )(f ⊗F∞ )i = ik−η ((k − 2)!)2 (Mf ⊗ Mψ−1 ) Q 1 p∈Σe (f ) (1 + p ) 2(Mψ−1 )(Af ) (bdR ⊗ιk−1 ), and b(Bf ) ∈ ∆(Bf ) by the formula b(Bf ) = (1 − k)(Af )tw(b(Af )). (1.1) Then L(Af , 0)(1 ⊗ b(Af )) = c+ (Af ) and L(Bf , 0)(1 ⊗ b(Bf )) = c+ (Bf ). We explain very briefly the content of this theorem below. Readers are advised to check the original paper for a full explanation. The fundamental line ∆(Af ) is identified with homK (Filk−1 Mf,dR ⊗ Q · F∞ , Mf,dR / Filk−1 Mf,dR ). There is also a perfect alternating pairing induced from the Poincaré duality for X1 (N ), h·, ·i : Mf ⊗Kf Mf → Mψ (1 − k), 9 where Mψ (1 − k) denotes the (1 − k)-Tate twist of the Dirichlet motive associated to ψ. F∞ is naturally viewed as a basis of A+ f,B . η = 0 or 1 so that η ≡ k mod 2. (M ) is the well-known epsilon factor appearing in the functional equation of a motive M defined in [De2] or [Ta, Theorem 3.4.1]. I p Σe (f ) is the set of primes p such that Mf,λ = 0 for any λ - p and Lp (Af , s) = (1 + p−s )−1 . A basis of Mψ,dR is bdR = X ψ(a) ⊗ e2πia/Nf ∈ OKf ⊗ OQ[e2πi/Nf ] )[1/Nf ], a∈(Z/Nf Z)× ι is the canonical basis of TdR , where T is the “dual of integral Tate motive.” Identifying ∆(Bf ) with homK (Filk−1 Mf,dR ⊗ Q(2)B , (Mf,dR / Filk−1 Mf,dR ) ⊗ Q(2)dR ), we define the isomorphism of K-lines tw : ∆(Af ) → ∆(Bf ) (1.2) so that tw(φ)(x ⊗ y) = φ(x ⊗ F∞ ) ⊗ β(y), where the basis β of ∆(Q(2)) sends (2πi)2 to ι−2 . This ends our explanation of Theorem 1.3.1. 10 The fundamental line ∆(A) is a free rank one T -module with basis b(A) = Y b(Af ). f ∈I We also have c+ (A) = Q f ∈I c + (Af ). In general, by taking the direct prod- uct over I, all computations in [D-F-G2, §8.2] immediately carry over from Af , Bf , Kf etc. to AI , BI , TI etc. Remark 1.3.2. Deligne’s conjecture predicts that L(Af , 0)−1 c+ (Af ) ∈ R ⊗ ∆(Af ) is an element in its T -rational subspace 1 ⊗ ∆(Af ) = ∆(Af ). The above theorem computes explicitly that this value is b(Af ) ∈ ∆(Af ). The same applied to Bf . Taking direct product over I, we obtain the later equality in (1.9). 1.4 The Burns-Flach conjecture in our case Our choice of integral version M of M comes from [D-F-G2]. Specifically, Mf is the integral premotivic structure associated to f defined in §5.4 of loc. cit. We construct M from all Mf , f ∈ I and then A and B from M naturally. Note that this choice is the same as in the typical example mentioned in §1.2. We denote by SI the set of prime numbers ` so that there exists an f ∈ I and λ ∈ Sf with λ | `. Here Sf is the finite set of primes λ in Kf such that either λ | N k! or the two-dimensional residual Galois representation Mf,λ /λMf,λ is not absolutely irreducible when restricted to GF , where 11 p F = Q( (−1)(`−1)/2 `) and λ | `. Clearly, we only expect to prove the BurnsFlach conjecture for primes ` ∈ / SI as our result will be built on the result of [D-F-G2, §8]. Recall M means MI . M` , a Galois stable lattice inside M` , is assumed to be free of rank two over the semilocal finite flat Z` -algebra T` := T ⊗Z Z` . We have B` = HomT` (M` , M` )0 (1) is a free T` -module of rank 3. Let D` := Dcrys (B` ) := H 0 (Q` , Bcrys,` ⊗Q` B` ), which is free of rank 3 over T` and tB ⊗Q Q` ∼ = tB` := D` / Fil0 D` , (1.3) which is free of rank 2 over T` . For each place v of Q Iv is a fixed choice of inertia group and φv is the geometric Frobenius. We define a perfect complex of T` -modules RΓf (Qv , B` ) :=    1−φv  H 0 (Iv , B` ) −−−→ H 0 (Iv , B` ) for v 6= `,   (1−φv ,π)  D` − −−−−→ D` ⊕ tB` for v = `. 12 So we have a canonical isomorphism of (graded) invertible T` -modules DetT` RΓf (Qv , B` )    0  ∼ DetT` H 0 (Iv , B` ) ⊗ Det−1 T` H (Iv , B` ) = T` :=    −1 −1 ∼ DetT` D` ⊗ Det−1 T` D` ⊗ DetT` tB` = DetT` tB` for v 6= ` (1.4) for v = `. For the usage of the determinant functor Det here, readers may see [K-M]. For the importance of keeping gradation data, see [B-F3, Remark 9]. As explained in [B-F3, §3.2] one can construct a natural morphism RΓf (Qv , B` ) → RΓ(Qv , B` ), (1.5) the mapping cone of which we denote by RΓ/f (Qv , B` ). The comparison isomorphism between etale and singular cohomology induces an isomorphism RΓ(R, B` ) ∼ = Q` ⊗Q BB+ . (1.6) Let Sbad be the set of places v of Q where M has bad reduction (i.e., where M`Iv 6= M` ) and put S = Sbad ∪ {`, ∞}. Define RΓf (Q, B` ) = Cone(RΓet (ZS , B` ) → M v∈Sbad ∪{`} RΓ/f (Qv , B` ))[−1]. (1.7) 13 Then there is an exact triangle (of perfect complexes of T` -modules) M RΓet,c (ZS , B` ) → RΓf (Q, B` ) → RΓf (Qv , B` ) ⊕ RΓ(R, B` ). (1.8) v∈Sbad ∪{`} By [B-F3, Lemma 19] and [D-F-G2, Theorem 8.2] the complex RΓf (Q, B` ) is acyclic. So the triangle (1.8) together with (1.3), (1.4) and (1.6) induces an isomorphism of (graded) determinants ϑ` : DetT` RΓet,c (ZS , B` ) ∼ = O −1 Det−1 T` RΓf (Qv , B` ) ⊗ DetT` RΓ(R, B` ) v∈Sbad ∪{`} + ∼ = DetT` (Q` ⊗Q tB ) ⊗ Det−1 T` (Q` ⊗Q BB ). Conjecture 1.4.1. (The `-primary part of Burns-Flach conjecture for the motive B with the action of T and ` ∈ / SI ) there is an identity of invertible T` -modules ϑ` (DetT` RΓet,c (ZS , B` )) = T` · b(B) = T` · L(B, 0)−1 c+ (B). (1.9) Remark 1.4.2. The conjecture can be formulated and expected to hold for all `. The restriction on ` is only to emphasize our inability to prove the excluded cases because our proof relies on [D-F-G2]. By its totality Burns-Flach conjecture means the Conjecture 4 in [B-F3], which consists of four parts. For motive B, part (i), (ii) and (iii) can be established readily. What we have (re)formulated here is actually the Conjecture 14 6 in loc. cit., whose validity for all primes is equivalent to part (iv). Remark 1.4.3. [F-P1] shows that the `-primary part of Bloch-Kato conjecture is equivalent to the equality ϑ` (DetOT` RΓet,c (ZS , B` ⊗T` OT` )) = OT` · L(B, 0)−1 c+ (B), which determines `-primary parts of L(Bf , 0)−1 · c+ (Bf ) up to multiplication by an element in (OKf ⊗ Z` )× for all f ∈ I once we decompose each sides into its components. The equality (1.9) further reduces the uncertainty of determining the `-primary parts of the special values by an algebraic formula, which is the main point of Burns-Flach conjecture compared to Bloch-Kato conjecture in this simplest non-trivial case. The exact extent to which this `-primary part of Burns-Flach conjecture for BI is finer than the combination of those of Bloch-Kato conjecture for all Bf , f ∈ I, is measured by (the cardinality of ) the finite group × O× T` /T` . 15 Chapter 2 An isomorphism between tangent space and Selmer group In the deformation theory of Galois representations there is a standard isomorphism between the relative Zariski tangent space of the universal deformation ring with a Selmer group, (i.e., a subspace of a global Galois cohomology group cut out by local conditions). This chapter is to prove in detail a variant of that isomorphism as shown in equality (2.1), which appears in [Ma2] with incomplete proof. This will be used in the next chapter. This variant, while of interest by itself, will be one of the critical steps in proving our case of Burns-Flach conjecture. 2.1 Notation Maps between objects that are naturally topological spaces are continuous. Galois groups are topological spaces with the canonical profinite topology. Any module that is a Z` -module by restriction of scalars is equipped with the `-adic topology. When we refer to an RG- module M , where M is a topological 16 space, R a ring and G a topological group, we mean an R-module M with an R-linear continuous G-action. H i (G, V ) is the continuous cochain cohomology group of G with coefficients in a continuous G-module V . First, we introduce the common setting of both sides of the isomorphism. Let GQ be the absolute Galois group after we fix an algebraic closure of Q. Fix a decomposition group Gp ⊂ GQ for all primes p in Z and Ip ⊂ Gp its inertia group. Let ` be a fixed odd prime so that our representation spaces will be `-adic and Σ a finite set of rational primes different from `. A is a commutative complete local ring that is finite free over Z` and whose residue field is k. Let L be a (continuous) AGQ -module that is free of rank two over A. Let FR be the full subcategory of finite Z` G` -modules whose objects are quotients of G` -stable lattices in short crystalline Q` representations of G` , where short representation V means that Fil` D(V ) = 0 and Fil0 D(V ) = D(V ), where D(V ) = (Bcrys,` ⊗Q` V )G` and that V has no nonzero subrepresentation V 0 so that V 0 (` − 1) is unramified. FR is stable under taking finite direct sums, subobject and quotients in the category of finite Z` G` -modules, and that it is equivalent to the category MF0tor that is defined in §5.1. We assume that L ⊗Z` Q` is a short crystalline representation. Now we introduce the objects related to the right hand side of the isomorphism (i.e., the Selmer group side). W = End0A (L) ⊗Z` Q` /Z` , where End0A (L) denotes the kernel of the trace map EndA (L) → A. Note that W is an AGQ -module as the trace map is 17 an AGQ -module homomorphism. Wn = W [`n ] = End0A (L) ⊗Z` (`−n Z` )/Z` = End0A (L/`n L) ⊂ EndA (L/`n L) where the last equality is given by f ⊗Z` (`−n mod Z` ) 7→ f ⊗Z` idZ` /`n ∈ EndA⊗Z` Z` (L ⊗Z` Z` /`n ) = EndA (L/`n L). Here we use that A is finite free over Z` . Note that W = ∪n Wn = limWn . W and Wn −→ for all n have the (natural) discrete topologies. For p 6= `, Hf1 (Gp , Wn ) = ker(H 1 (Gp , Wn ) → H 1 (Ip , Wn )). For p = `, recall there is a canonical A/`n -linear isomorphism between H 1 (G` , EndA (L/`n L)) and the A/`n -module of Yoneda extensions of A/`n G` -modules 0 → L/`n L → E → L/`n L → 0 as L/`n L is a free A/`n -module. Hf1 (G` , Wn ) ⊂ H 1 (G` , Wn ) is the set of elements that corresponds to extensions E in FR when mapped into H 1 (G` , EndA (L/`n L)). One checks that Hf1 (G` , Wn ) is an A-submodule of H 1 (G` , Wn ). Hf1 (Gp , W ) = limHf1 (Gp , Wn ) for every prime p. In particular, −→ for p 6= `, Hf1 (Gp , W ) = ker(H 1 (Gp , W ) → H 1 (Ip , W )) = H 1 (GFp , W Ip ). We note that limH 1 (Gp , Wn ) = H 1 (Gp , W ) because of the compactness −→ of Gp and the discreteness of the topology of W . So Hf1 (Gp , W ) is a subset of H 1 (Gp , W ). HΣ1 (GQ , W ) ⊂ H 1 (GQ , W ) is the set of elements that are in Hf1 (Gp , W ) when restricted to H 1 (Gp , W ) for every p 6∈ Σ . HΣ1 (GQ , Wn ) ⊂ H 1 (GQ , Wn ) is the set of elements that are in Hf1 (Gp , Wn ) when restricted to H 1 (Gp , Wn ) for every p 6∈ Σ. All of above cohomology groups are naturally A-modules. The Selmer groups HΣ1 (GQ , W ) and HΣ1 (GQ , Wn ) depend on L as a GQ -module, not on the choice of Gp of any p. 18 Finally, we come to the setting of the left hand side of the isomorphism (i.e., the deformation theory side). Let k be the finite residue field of A whose characteristic is `. Fix a basis of L over A so that the action of GQ on L has a matrix representation ρ : GQ → GL2 (A) and its residual matrix representation ρ̄ : GQ → GL2 (k). Let O be a complete local noetherian ring with residue field k. Let CO = C denote the category whose objects are complete local noetherian O-algebras and whose morphisms are local O-algebra homomorphisms. Hence k and O are objects of C. There is a canonical map from Z` to any object of C. We require that A is an object in C and hence that O has the same characteristic 0 as A. For example, we can take O to be A, or the canonical subring W (k) of A, which is the ring of Witt vectors with coefficients in k. Recall that if R is an object of C, then an R-deformation of ρ̄ is a strict equivalence class of liftings of ρ̄ to GL2 (R) [Ma3, §8]. We say that an R-deformation is of type Σ if a lifting representing that deformation, whose representation space is M , satisfies the following conditions, which will be referred to individually by the phrases in the parenthesis: • the RGQ -module M is minimally ramified outside Σ∪{`}, where minimal 19 ramification is defined in [Di], (minimal ramification); • for every n > 0, the Z` G` -module M/Mn M is an object of the category FR, where M is the maximal ideal of R, (crystalline ramification); • ∧2R M is the one dimensional representation over R whose character is the `-cyclotomic character composed with the canonical map from Z` to R, (fixed determinant). Assume ρ̄ itself is a deformation of type Σ so that we can consider the functor DΣ on C that associates to R the set of R-deformations of ρ̄ of type Σ. Assume further that ρ̄ is absolutely irreducible. By the results of Mazur [Ma3] and Ramakrishna [Ra], this functor is representable by an object of C that is called the universal deformation ring attached to ρ̄ with base ring O of type Σ. We denote this object Rρ,O,Σ or simply RΣ . If ρ is a deformation of type Σ, the universality of RΣ induces a ring homomorphism θρΣ : RΣ → A, by which A becomes an RΣ -algebra. 2.2 The isomorphism The standard isomorphism mentioned in the beginning of this chapter is, in our context, of the form homA (ΩRΣ /O ⊗RΣ A, A) = HΣ1 (GQ , End0A (L)), where the right hand side is a suitable subset of H 1 (GQ , End0A (L)). However, what we will need is the following variant: Proposition 2.2.1. With the notation and assumptions of previous para- 20 graphs, we have a canonical A-module isomorphism: homA (ΩRΣ /O ⊗RΣ A, A ⊗Z` Q` /Z` ) ∼ = HΣ1 (GQ , W ), (2.1) where ΩRΣ /O is the topological RΣ -module of relative continuous Kähler differentials of O-algebra RΣ that represents the functor M 7→ DerO (RΣ , M ) for a topological RΣ -module M . Proof. It suffices to show the following three claims. • HΣ1 (GQ , W ) ∼ = limHΣ1 (GQ , Wn ). −→ • homA (ΩRΣ /O ⊗RΣ A, A ⊗Z` Q` /Z` ) ∼ = lim homA (ΩRΣ /O ⊗RΣ A, `−n A/A), −→ where `−n A/A is a shorthand for A ⊗Z` `−n Z` /Z` . • There exists a canonical A-module isomorphism homA (ΩRΣ /O ⊗RΣ A, `−n A/A) → HΣ1 (GQ , Wn ) that makes the following diagram commute, homA (ΩRΣ /O ⊗RΣ A, `−n A/A) ↓ → HΣ1 (GQ , Wn ) ↓ homA (ΩRΣ /O ⊗RΣ A, `−n−1 A/A) → HΣ1 (GQ , Wn+1 ). Proof of the first claim. 21 There is a natural map from the right hand side to the left hand side induced by the inclusion Wn → W . That map is injective because of the compactness of GQ and the discreteness of the topology of W since a (continuous) map from a compact space to a discrete space has only finitely many values. Now we prove surjectivity. Take any c̄ ∈ HΣ1 (GQ , W ), which is represented by a cocycle c : GQ → W , whose restriction to Gp , name it cp , represents an element in Hf1 (Gp , W ) = limHf1 (Gp , Wn ) for all p 6∈ Σ. As no confusion can −→ arise, we will also use the same symbol for a cocycle if we restrict its domain and/or its codomain to a smaller one. As we have just said, c has only finitely many values. Suppose these values are in Wm for some m. By the deformation condition of minimal ramification, ρ is unramified at a prime when so is ρ̄. Since the image of ρ̄ is finite, ρ̄ is unramified outside a set S of finitely many primes and hence so is ρ. For all primes p 6∈ S ∪ Σ, Hf1 (Gp , W ) = lim ker(H 1 (Gp , Wn ) → H 1 (Ip , Wn )) −→ = ker(H 1 (Gp , W ) → H 1 (Ip , W )), where H 1 (Ip , W ) = hom(Ip , W ) as Ip acts trivially on W . So, c is trivial on Ip . Hence cp represents an element in Hf1 (Gp , Wn ) for all n. For all primes p in the finite set S − Σ, cp represents an element in Hf1 (Gp , Wn0 ) simultaneously for a n0 ≥ m that is large enough. Hence, c also represents an element in HΣ1 (GQ , Wn0 ). This means the surjectivity we want. Proof of the second claim. 22 Since A ⊗Z` Q` /Z` = ∪n `−n A/A, it suffices to prove ΩRΣ /O ⊗RΣ A is a finitely generated A-module, or, equivalently, ΩRΣ /O is a finitely generated RΣ -module. By Cohen structure theorem on complete local Noetherian rings of unequal characteristics as discussed after [Ei, Theorem 7.8] we have RΣ ∼ = T /I, where T = W (k)[[x1 , · · · , xr ]] for an integer r, I an ideal of T. Recall there is a canonical homomorphism from W (k) to any complete local ring with residue field k, which manifests in the isomorphism above. Thus O can be viewed as a sub-W (k)-algebra of RΣ via the structure map O → RΣ . We have two canonical RΣ -module surjections: ˆ T ΩT /W (k) → ΩRΣ /W (k) → ΩRΣ /O , RΣ ⊗ ˆ denotes completed tensor product. See §5.2.3 of [Hi2]. Since ΩT /W (k) where ⊗ is a free T -module of rank r, ΩRΣ /O is a finitely generated RΣ -module. Proof of the third claim. 23 The method is explained partially by B. Mazur in [Ma2]. We have homA (ΩRΣ /O ⊗RΣ A, `−n A/A) = homRΣ (ΩRΣ /O , homA (A, `−n A/A)) = homRΣ (ΩRΣ /O , A/`n ) = DerO (RΣ , A/`n ) = {φ ∈ homO−algebra (RΣ , A/`n []) : φ(r) ≡ (θρΣ (r) mod `n ) mod } = {ρ-relative deformation of ρ̄ to A/`n [] of type Σ}, where A/`n [] = A/`n [X]/(X 2 ) with  = X mod (X 2 ) so A/`n [] is an object in C with the square zero idea (). Also the second equality from the bottom is given by d 7→ (r 7→ (θρs (r) mod `n + d(r))). From now on, we view A/`n as a subset of A/`n [] and GL2 (A/`n ) as a subset of GL2 (A/`n []) in the natural way whenever necessary. We say a lifting of ρ̄ to A/`n [] is ρ-relative if its reduction module () is ρL mod `n . A deformation of ρ̄ to A/`n [] is said to be ρ-relative if it is represented by a ρ-relative lifting of ρ̄. We will produce a bijection from the set in the last row of above array, name it tρ,n , to HΣ1 (GQ , Wn ). Let θ : GQ → GL2 (A/`n []) be a ρ-relative lifting of ρ̄. There is a canonical such lifting, call it θ0 : namely, the composition of ρL mod `n and the nat- ural embedding GL2 (A/`n ) ,→ GL2 (A/`n []). We associate to θ the difference cocycle cθ : GQ → M2 (A/`n ) ∼ = EndA (L/`n L) 24 by 1 +  · cθ (g) = θ(g) · θ0 (g)−1 for g ∈ GQ , where M2 (A/`n ) denotes the underlying additive group of the A/`n -algebra of 2 × 2 matrices with entries in A/`n and where the latter isomorphism is induced by the fixed basis of L. cθ is well defined, i.e., θ(g) · θ0 (g)−1 ∈ 1 +  · M2 (A/`n ), because θ is ρrelative. This construction is independent of the choice of the basis of L. In fact, we can make each step of the construction coordinate free. Check that this construction provides a bijection between the set of ρ-relative liftings of ρ̄ to A/`n [], and the set Z 1 (GQ , EndA (L/`n L)) of 1-cocycles. Then check that under this bijection, two liftings are strictly equivalent if and only if their associated cocycles are cohomologous. A straightforward reduction of definitions shows that θ satisfies the deformation condition of fixed determinant if and only if the values of cθ reside in Wn = End0A (L/`n L), a direct summand of AGQ -module EndA (L/`n L). Combining with the previous paragraph, we see that the map the A/`n []-deformation class of θ 7→ the cohomology class of cθ (2.2) is a bijection between the set of ρ-relative A/`n []-deformations of ρ̄ with fixed determinant and H 1 (GQ , Wn ). Now we will check that θ is minimally ramified at any prime p outside Σ and different from ` if and only if the restrictions of θ and θ0 to Ip can be brought one into another by conjugation by elements in the kernel of reduction GL2 (A/`n []) → GL2 (A/`n ), which in turn if and only if the cocycle cθ is co- 25 homologous to zero when restricted to Ip ⊂ Gp . A straightforward calculation shows the second “if and only if.” Now we will prove the first “if and only if.” There are three cases according to the ramification of ρ̄. In each case, the “if” part is trivial. So we will write only the proof of “only if” part, assuming that θ is minimally ramified at a prime p outside Σ and different from `. We use e to denote composition with the Teichmüller lift k × → W (k)× → A/`n , (or its further composition with A/`n ,→ A/`n []) as in [Di], where all three cases of minimal ramification  are given.  ξ1 0   . Then after a suitable choice of basis of Vθ0 and Case 1. ρ̄|Ip ∼    0 ξ2   ξ˜1 0   and a corresponding choice of basis of Vθ , we may assume θ0 |Ip =    0 ξ˜2     m 1 m 2  ξ˜1 0   : Ip → M2 (A/`n ).  for a map m =  θ|Ip = θ0 +  · m ∼      0 ξ˜2 m3 m4 Assume ξ˜1 6= ξ˜2 , otherwise θ0 |Ip = θ|Ip already. We know   ξ˜1 0  , (a + b)θ|Ip (a + b)−1 =    0 ξ˜2   26    a1 a2  b 1 b 2  , b =   ∈ M2 (A/`n ). Expanding the equality, for some a =      a3 a4 b3 b4 we find it is equivalent to       0 m1 m2    a1 0  =  , and  a=      ˜ ˜ b3 a−1 m3 m4 0 a4 1 (ξ2 − ξ1 )  .  0    Hence θ|Ip is brought to θ0 |Ip by conjugation by   ˜ ˜ b2 a−1 4 (ξ1 − ξ2 ) 1 b3 a−1 1  · b2 a−1 4    1  1 ∗  . Then after suitable choices of bases, we may Case 2. ρ̄|Ip ∼ ξ ⊗    0 1     1 ∗  1 ∗  and θ|Ip = θ0 +  · m ∼ ξ˜⊗  . After twisting assume θ0 |Ip = ξ˜⊗      0 1 0 1 θ and θ0 with ξ˜−1 , this case is explained in [Ma3, §29]. Case 3. ρ̄|Ip ∼ IndIIPM ξ, where M is a ramified quadratic extension of Qp and ξ is a character of IM that is not equal to its conjugate ξ 0 under the action of a lift σ of the nontrivial element in Gal(M/Qp ) to Ip . Note that Ip= IM ∪ σIM . ξ˜ 0   and Then after suitable choices of bases, we may assume θ0 |IM =    0 ξ˜0   0 1 ˜ So, for  as θo |Ip ∼ IndIP ξ˜ and that θ|Ip = θ0 +  · m ∼ IndIP ξ. θ0 (σ) =  IM IM   1 0     0 1 ξ˜ 0  n −1  and aθ(σ)a−1 =   . Write some a ∈ GL(A/` []), aθ|IM a =      0 ξ˜0 1 0 ā for the reduction of a to GL2 (A/`n ). Then āθ0 (τ )ā−1 = θ0 (τ ) for τ ∈ Ip . 27 Check that this implies ā is a scalar matrix. Hence, θ|Ip is brought to θ0 |Ip by conjugation by ā−1 a, which reduces to the identity matrix of GL(A/`n ). Now we deal with the deformation condition of crystalline ramification at `. We naturally identify the representation space of θ, Vθ with A/`n -module L/`n L ⊕  · L/`n L as θ is a ρ-relative lifting. Then we have an extension of (A/`n )GQ -modules 0 → ·L/`n L → Vθ → L/`n L → 0, which, by the canonical isomorphism mentioned when defining Hf1 (G` , Wn ), corresponds to the elements in H 1 (G` , Wn ) represented by cocycle c0θ given by  ·(c0θ (g)(e)) = θ(g)(α(θ(g)−1 e)) − e for g ∈ GQ and e ∈ L/`n L, where α : L/`n L → Vθ is any fixed A/`n -module splitting of the above extension. Choosing α to be the natural splitting, we find that c0θ is none other than cθ restricted to G` . [La, Chapter 10, Corollary 5.7], or the Krull intersection theorem as stated in [Ei, Corollary 5.4], tells that the maximal ideal of the finite local ring A/`n [] is nilpotent. So θ satisfies the deformation condition of crystalline ramification if and only if Vθ is an object in FR, i.e., if and only if the cocycle cθ restricted to G` represents an element in Hf1 (G` , Wn ). Just tautology. Pulling together the conclusions of above paragraphs on local ramification 28 restrictions and the definition of HΣ1 (GQ , Wn ), we see that the map (2.2) is a bijection between tρ,n and HΣ1 (GQ , Wn ). So we have constructed a bijection between homA (ΩRΣ /O ⊗RΣ A, Q` /Z` ) and HΣ1 (GQ , Wn ). It is straightforward to verify that the bijection is actually an A-module isomorphism. It is much more tedious to check the diagram in the third claim is commutative. Remark: The isomorphism is functorial in an obvious way if we shrink the set Σ. Remark: Suppose we choose O to be A. Let η be the kernel of θρΣ : RΣ,A → A. Then RΣ = A ⊕ η as A-module. We have a topological A-module isomorphism ΩRΣ /O ⊗RΣ A = η/η 2 . This is proved by showing directly that the obvious homomorphism from the left hand side to the right hand side is an isomorphism. 29 Chapter 3 Reduction and proof of ad0(H 1(N ))(1) We shall focus on BI rather than AI because the methods of [D-F-G2] apply more directly to BI (see in particular its Lemma 8.11). Once we proved the case on BI , the desired result on AI can be formulated similarly and can be seen along the same line of arguments. 3.1 A reformulation in classical terms In this section we transform the left equality of (1.9) into a concrete identity of elements in T`× /T× ` . In order to do that, or what amount to the same thing, reformulate the Burns-Flach conjecture 1.4.1 in classical terms such as the Fitting ideal of HΣ1 , one needs to construct the triangle (1.8) with B` replaced by B` . Assumption: The submodule Fil0 (B`-crys ) is a T` -direct summand of B`-crys . Readers are cautioned that a general motive does not determine its integral version(s) in any canonical way. We can use the notation B, B`-crys and B` 30 as we have specified B explicitly in §1.4. In our case of M = H 1 (X0 (N )) we know by Mazur’s results [Ma1] that Fil0 M`-crys ∼ = H 0 (X0 (N )/Z` , Ω1 ) is a free T` -module of rank 1. Since T` is Gorenstein the Z` - dual M`-crys / Fil0 ∼ = H 1 (X0 (N )/Z` , O) is also free of rank 1 over T` and hence M`-crys is free over T` . By linear algebra the Fili (B`-crys ) = Fili HomT` (M`-crys , M`-crys (1))0 is then also free over T` . The assumption still needs to be verified, however, in more general situations. Ideally the only thing we’d like to assume is that M` is free over T` . Proposition 5.1.2 implies M`-crys is free over T` if V(M`-crys ) ∼ = M` and Proposition 5.1.1 and remark 5.1.3 imply that Fili M`-crys is also free over T` . Under the assumption we can define a perfect complex RΓf (Q` , B` ) of T` modules by 1−φ0 RΓf (Q` , B` ) = Fil0 B`-crys −−−→ B`-crys and a map (in the derived category) RΓf (Q` , B` ) → RΓ(Q` , B` ) as follows. There is a commutative diagram of GQ` -modules (with B`-crys having trivial action) Fil0 B`-crys   y 1−φ0 −−−→ B`-crys   y 1⊗(1−φ0 ) B` ⊗Z` Fil0 Acrys,` [−k](−k) −−−−−→ B` ⊗Z` Acrys,` [−k](−k) k B` ⊗Z` Filk Acrys,` (−k) (3.1) k 1⊗(1−p−k φ) −−−−−−−→ B` ⊗Z` Acrys,` (−k) where [k] (resp. (k)) denotes filtration shift (resp. Tate twist) and Acrys,` is 31 the ring defined by Fontaine. The lower row is a resolution of the GQ` -module B` obtained by tensoring the exact sequence [B-K, (2.5.1)] 1−p−k φ 0 → Zp (k) → Filk Acrys,` −−−−→ Acrys,` → 0, with B` (−k). Denoting the lower row in (3.1) by E • (B` ) and the standard continuous cochain resolution by C • (GQ` , −) we have maps (3.1) RΓf (Q` , B` ) −−→ E • (Bλ ) → C • (GQ` , E • (B` )) ← C • (GQ` , B` ) = RΓ(Q` , B` ) as desired. Tensoring with Q` , we obtain the map RΓf (Q` , B` ) → RΓ(Q` , B` ) defined in [B-F3, §3.2]. We are lucky that the complex RΓf (Q` , B` ) already turns out to be perfect over T` . This is not necessarily the case for the complex 1−φv RΓf (Qv , B` ) = H 0 (Iv , B` ) −−−→ H 0 (Iv , B` ) for v ∈ Sbad since the submodule H 0 (Iv , B` ) of B` is not always T` -free. Therefore when defining the global complex RΓf (Q, B` ) we do not follow (1.7) but we choose a set Σ ⊆ Sbad so that RΓf (Qv , B` ) is T` -perfect for v ∈ / Σ and 32 define RΓf ,Σ (Q, B` ) = Cone(RΓet (ZS , B` ) → M RΓ/f (Q` , B` ) ⊕ v∈Σ0 M RΓ(Qv , B` )[−1] v∈Σ (3.2) where Σ0 = Sbad ∪ {`} \ Σ. With this definition there is an exact triangle of perfect complexes of T` -modules RΓet,c (ZS , B` ) → RΓf ,Σ (Q, B` ) → M RΓf (Qv , B` ) ⊕ RΓ(R, B` ). (3.3) v∈Σ0 and a map of triangles of perfect complexes of B` -modules RΓet,c (ZS , B` ) −−−→ RΓf ,Σ (Q, B` ) −−−→     y y RΓet,c (ZS , B` ) −−−→ RΓf (Q, B` ) −−−→ L v∈Σ0 RΓf (Qv , B` ) ⊕ RΓ(R, B` )   y L RΓf (Qv , B` ) ⊕ RΓ(R, B` ), v∈Sbad ∪{`} (3.4) where the upper row is (3.3) tensored with Q` . If ω is a T -basis of detT (tB ) and h a T -basis of detT BB+ , then h−1 ⊗ ω is a T -basis of ∆(B). If b is any other T -basis of ∆(B) we denote by b · h ⊗ ω −1 the unique scalar λ ∈ T × so that b = λ · h−1 ⊗ ω. Denote by W ∨ = Homcont (W, Q` /Z` ) the Pontryagin dual of a profinite or discrete Z` module W . Lemma 3.1.1. Let ω be T -basis of detT (tB ) that is also a T` -basis of the T` lattice detT` (B`-crys / Fil0 B`-crys ), and let h be a T -basis of detT BB+ that is also 33 a T` -basis of detT` (B+ ` ). Put LΣ (B, 0) = L(B, 0) Y Lv (B, 0)−1 v∈Σ and bΣ (B) = LΣ (B, 0)−1 c+ (B). For any prime ` ∈ / SI the Burns-Flach conjecture 1.4.1 is equivalent to the identity T` · bΣ (B) · h ⊗ ω −1 = #HΣ1 (Q, A` /A` )∨ , (3.5) where #W denotes the Fitting ideal of a finite T` -module W of finite projective dimension (i.e., the class of W in the relative algebraic K-group K0 (T` , Q` ) ∼ = T`× /T× ` ). Proof. Noting that the vertical maps in (3.4) are quasi-isomorphisms we obtain 34 a commutative diagram of isomorphisms of (graded) invertible T` -modules DetT` RΓet,c (ZS , B` )   y DetT` RΓet,c (ZS , B` )   y DetT` RΓf (Q, B` )⊗ O Det−1 T` RΓf (Qv , B` )⊗ DetT` RΓf ,Σ (Q, B` )⊗ α −−−→ O Det−1 T` RΓf (Qv , B` )⊗ v∈Σ0 v∈Sbad ∪{`} (3.6) Det−1 T` RΓ(R, B` ) Det−1 T` RΓ(R, B` )     y βy DetT` (Q` ⊗Q tB ) Q v∈Σ Lv (B,0) DetT` (Q` ⊗Q tB ) −−−−−−−→ + ⊗ Det−1 T` (Q` ⊗Q BB ) + ⊗ Det−1 T` (Q` ⊗Q BB ) where the left hand vertical map is ϑ` and all unspecified tensor product are over T` . The factor Lv (B, 0) appears because for v ∈ Σ the isomorphism DetT` RΓf (Qv , B` ) ∼ = T` (3.7) given in (1.4) (which is used for β) differs from the isomorphism (3.7) induced by the quasi-isomorphism RΓf (Qv , B` ) → 0 (which is used for α) by precisely this factor (see [B-F2, Lemma 1]). So conjecture 1.4.1 is equivalent to Y v∈Σ Lv (B, 0)ϑ` (DetT` RΓet,c (ZS , B` )) = T` · bΣ (B) = T` · LΣ (B, 0)−1 c+ (B). 35 Using the commutativity of (3.6) together with (3.3) this is equivalent to Σ DetT` RΓf ,Σ (Q, B` ) ⊗ DetT−1` RΓf (Q` , B` ) ⊗ Det−1 T` RΓ(R, B` ) = T` · b (B) or DetT` RΓf ,Σ (Q, B` ) = T` · bΣ (B) · h ⊗ ω −1 . It remains to compute the cohomology of RΓf ,Σ (Q, B` ) and this can be done along the lines of [B-F1, (1.35)-(1.37)]. We have Hf0,Σ (Q, B` ) = H 0 (Q, B` ) = 0, where the last equality holds as Mf,λ is absolutely irreducible and not isomorphic to Mf,λ (1) for λ | ` a place of Kf . We also have Hf3,Σ (Q, B` ) ∼ = H 0 (Q, A` /A` )∨ = 0, where the last equality is equivalent to the assumption on ` that Mf,λ /λMf,λ is absolutely irreducible when restricted to GQ√(−1)(`−1)/2 ` . Moreover we know Hf1,Σ (Q, B` ) = Hf1 (Q, B` ) = 0 and hence Hf1,Σ (Q, B` ) = H 0 (Q, B` /B` ) = 0 since ` ∈ / SI . Finally using Tate-Poitou duality, we have an exact sequence 0 → Hf2,Σ (Q, B` )∨ → H 1 (ZS , A` /A` ) → M H 1 (Qv , A` /A` ) v∈Σ0 Hf1 (Qv , A` /A` ) , 36 which identifies Hf2,Σ (Q, B` ) with HΣ1 (Q, A` /A` )∨ . 3.2 Reduction to a concrete problem From now on we shall focus on our case of M = H 1 (X0 (N )) with N prime, since we can not prove the conjecture in its full generality. We recall the restriction on `: ` 6∈ SI , which enable us to use many facts in [D-F-G2]. We note that the reformulation in the previous section is valid. Let T` = Q m Tm be the decomposition of T` into complete local O` - algebras, where m runs through all maximal ideals of T containing `. For Σ each m we denote by Rm the universal deformation ring attached to the Galois representation Mf,λ /λMf,λ over the finite field Tm /m of type Σ, where f is a newform associated to the maximal ideal m of T (that is, m is the kernel of the algebra homomorphism T → C, t 7→ a1 (t(f ))) and λ is the place of Kf whose valuation ring contains m ∩ Kf . Note Mf,λ is short crystalline as ` - 2N . Put R`Σ = Q Σ Σ m Rm , so that there is a natural surjection R` → T` that comes from Σ the surjection Rm → Tm for all m. We know by equality (2.1) that there is a canonical isomorphism of T` modules HomT` (ΩR`Σ /Z` ⊗R`Σ T` , T` /T` ) ∼ = HΣ1 (Q, A` /A` ). The powerful method of Diamond-Taylor-Wiles that is implemented in [D-F-G2, Σ is a local complete intersection over Tm . So §7.2 and §7.3] leads to the fact Rm 37 for each m there is an integer rm and power series g1 , ..., grm ∈ Z` [[X1 , ...Xrm ]] such that Σ ∼ Rm = Z` [[X1 , ...Xrm ]]/(g1 , ..., grm ). In particular, ΩR`Σ /Z` ⊗R`Σ T` is a finite T` -module of finite projective dimension whose Fitting ideal is generated by the element ∆Σ = π det where π : Q m Z` [[X1 , ...Xrm ]] → Q  ∂gi ∂Xj !  , 1≤i,j≤rm Σ Σ m Rm = R` → Q m m Tm = T` is the composite ring homomorphism. Assuming that T` is Gorenstein we have HΣ1 (Q, A` /A` )∨ ∼ = ΩR`Σ /Z` ⊗R`Σ T` so that we need to show T` · bΣ (B) · h ⊗ ω −1 = T` · ∆Σ . The restriction on ` at the beginning of this chapter ensure us to use the explanation in [D-F-G2, §8.2] so that we know the element bΣ (B) is uniquely 38 determined by the scalar λ(bΣ (B)) ∈ T × such that < f, bΣ (B)(f ⊗ (2πi)2 ) ⊗ ι2 >= λ(bΣ (B)) · bdr ⊗ ι1 and similarly for h ⊗ ω −1 . Hence we have bΣ (B) · h ⊗ ω −1 = λ(bΣ (B)) . λ(h ⊗ ω −1 ) Note that we are only interested in λ(bΣ (B)) up to factors in T× ` , hence we can forget the factor ik−η ((k − 2)!)2 /2 occurring in Theorem 8.5. This factor × is the same for all f ∈ I hence lies in the diagonally embedded Z× ` ⊂ T` . In our case of M = H 1 (X0 (N )) with N prime, we can take Σ = ∅ because we have Lemma 3.2.1. RΓf (Qv , B` ) is T` -perfect for all v - `. Proof. This is clear if v - N as then M` is unramified due to the good reduction of X0 (N ) at primes other than N . Now suppose v = N . Then then action of Iv on the M` is unipotent and 39 T` -equivariant but nontrivial on any constituent of M` . Then BI`N = HomT` (M` , M` (1))IN    a = {  c   a = {  c   0 = {  0  b   a  ∈ M2 (T` ) |    −a c  b    1   −a 0 φ(σ)   1  =    1 0 b   φ(σ)  a   1 c  b −a  }    ∈ M2 (T` ) | φ(σ)a = 0, φ(σ)c = 0}  −a  b   ∈ M2 (T` )},  0   1 where σ runs through IN , acting on M` by   0  φ(σ)  . Hence BIN is a free T`  ` 1 module. We also know that Mf,λ for f ∈ I is minimally ramified at all maximal ideals of OKf not containing `. From now on, we replace Σ by ∅ or do not write it at all. By the results of Chapter 7 of [D-F-G2] the map R`∅ → T` is an isomorphism. Now let us look at Theorem 1.3.1 for each f ∈ I. We have k = 2, η = 0. Moreover, the character ψ for f is trivial so we can forget the degenerated Dirichlet motive Mψ−1 = Kf . There is no exceptional primes as we know an exceptional prime p must be N since p must divide the level prime number N . However, [Hi3, Theorem 4.2.4] says that decomposition group at p acting on Mf,λ for a prime λ | ` in Kf is equivalent to a diagonal representation:   ∗  η(σ)χ` (σ)  σ 7→    0 η(σ), 40 where χ` : GQ → Z× ` is the `-adic cyclotomic character and η is an unramified I p character. Hence the dimension of Mf,λ is not zero so p is not exceptional. (Mf ) equals ±N whereas (Af ) = N 2 . The sign of (Mf ) depends on f . It coincides, however, with the eigenvalue of the endomorphism of J0 (N ) on f induced by the conjugation on Γ0 (N ) by wN = ( N0 −1 0 ) . Hence the tuple of (Mf ) for f ∈ I is an element of T× by [Ri, Corollary 3.3]. So we can forget all factors in Theorem 1.3.1, where we have trivial character ψ and no exceptional primes, in the sense that λ(b(B)) ∈ T× ` . In order to make further progress, we also fix a number field K large enough to contain all complex embeddings of all fields Kf for f ∈ I and replace M Q by M ⊗ K, T by T ⊗Q K ∼ = IT K, T by T ⊗Z O and A by A ⊗ K. Here O denotes the ring of integers of K. Then we have T` = T` ∩ (T` ⊗Z` O` ). By Theorem 4.1 and Lemma 11 of [B-F3] Burns-Flach conjecture for this scalar extended motive implies the one for the original motive. Now we analyze λ(h ⊗ ω −1 ). The invertible T` -module detT` (B+ ` ) can be analyzed as in [D-F-G2, §8.2] and we find that the isomorphism detT BB+ ∼ = ∼ T (2)B can be chosen to induce an isomorphism detT` (B+ ` ) = T` (2) under which h maps to (2πi)2 . The isomorphism of [D-F-G2, equation (43)] det tB ∼ = HomT (Fil1 MdR , MdR / Fil1 ) ⊗Q Q(2)dR T 41 likewise induces an isomorphism detT` (B`-crys / Fil0 B`-crys ) ∼ = HomT` (Fil1 M`-crys , M`-crys / Fil1 ) ⊗Z` Z(2)`-crys . Hence we can view ω ⊗ ι2 as a basis of HomT` (Fil1 M`-crys , M`-crys / Fil1 ). Now note that the pairing <, > is induced by a perfect (Poincare duality) pairing on M. In particular it induces isomorphisms of free rank 1 T` -modules M` ∼ = HomO` (M` , O` ) (see the axioms in [D-F-G2, §7.2]) and M`-crys ∼ = HomO` (M`-crys , O` ). Note that T` is a finite flat Gorenstein O` -algebra contained in the maximal O` -order Y O` IT of T` . By definition λ(h ⊗ ω −1 ) is the element (λτ )τ ∈IT of this product given by < f τ , (h ⊗ ω −1 )(f τ ⊗ (2πi)2 ) ⊗ ι2 >=< f τ , (ω ⊗ ι2 )(f τ ) >= λτ · bdR ⊗ ι, where we view τ as an embedding Kf → K. 42 3.3 Differ by a unit in T` At this point we have nearly reduced the problem to an algebraic one. We are given • A local complete intersection R`∅ ∼ = T` finite flat over O` so that T` ⊗Z` Q` ∼ = Q I T K` . • A free T` -module M`-crys of rank two with a perfect O` -linear T` -balanced alternating pairing <, >. M`-crys contains a totally isotropic free rank 1 T` -submodule Fil1 M`-crys so that M`-crys / Fil1 is also T` -free. Hence we can pick a T` -isomorphism ω : Fil1 M`-crys → M`-crys / Fil1 . • For each τ ∈ IT we are given a O` -generator f τ of {x ∈ Fil1 M`-crys |tx = πτ (t)x} (the πτ -eigenspace) where πτ : T` → O` is the O` -algebra homomorphism given by projection onto the component indexed by τ ∈ IT . • Some knowledge of the modular forms, especially that all f τ are newforms, i.e., having first Fourier coefficient equal to 1. The final problem: Show that the elements Y
< f τ , ω(f τ ) > τ ∈ O` IT 43 and ∆∅ ∈ T` ⊂ Y O` IT
τ τ differ by an element in T× ` (in particular this implies that < f , ω(f ) > τ lies in T` ). Note that this problem does not depend on the choice of ω. Any other choice is of the form λω with λ ∈ T× ` and we have < f τ , λω(f τ ) >=< f τ , ω(λ·f τ ) >=< f τ , ω(πτ (λ)f τ ) >= πτ (λ) < f τ , ω(f τ ) > .
Hence we only change the element < f τ , ω(f τ ) > τ by λ ∈ T× ` . In any case, the problem is now sufficiently concrete. We exploit the fact that T` , being a complete intersection over O` , is Gorenstein. In particular one can choose a Gorenstein trace φ : T` → O` , i.e., a T` -basis of HomO` (T` , O` ). This choice induces an isomorphism HomO` (M`-crys , O` ) ∼ = HomT` (M`-crys , T` ), in other words there is a unique T` -bilinear pairing << −, − >> on M`-crys so that φ(<< x, y >>) =< x, y >. Since πτ are O` -linear there are also unique elements eτ ∈ T` so that πτ (x) = φ(eτ · x) for all x ∈ T` . Proof of the final problem: Since φ is a T` -basis of HomO` (T` , O` ), we can define a T` -isomorphism γ : HomO` (T` , O` ) → T` so that tφ 7→ t for t ∈ T` . 44 From [M-R, Appendix A.13], we know that there exists some element µ in T× ` so that the first of the following equalities holds. (µ∆∅ )φ = trT` /O` = X πτ = τ ∈IT X τ ∈IT eτ φ = ( X eτ )φ, τ ∈IT where eτ = γ(πτ ). (The second equality can be explained if we realize that trT` /O` = trT` ⊗K/K |T` and T` ⊗ K = Q IT K). Since γ is an isomorphism, we have µ∆∅ = X eτ . τ ∈IT We have a T` -module isomorphism κ : Fil1 M`-crys = S2 (Γ0 (N ), O` ) → HomO` (T` , O` ) defined by κ(f ) = (t 7→ (a1 (T f ))). See [D-I, §12.3], [D-D-T, lemma 1.34 ] or [Hi2, Theorem 3.17]. Proposition 3.3.1. κ−1 (πτ ) = f τ . Proof. : for all t ∈ T` , t(κ−1 (πτ )) = κ−1 (t(πτ )) = κ−1 (πτ (t)πτ ) = πτ (t)κ−1 (πτ ), 45 where the middle equality is justified by t(πτ )(x) = πτ (tx) = πτ (x)πτ (x) = (πτ (x)πτ )(x). So k −1 (πτ ) is in the πτ -eigenspace of Fil1 M`-crys , which is generated by f τ . So k −1 (πτ ) = cf τ for some c ∈ O` . Then c = a1 (cf τ ) = a1 (1 · (cf τ )) = κ(cf τ )(1) = πτ (1) = 1, where in the first equality we use the fact that f τ is a newform, i.e., its first Fourier coefficient is equal to 1. Other equalities are formal. Now note the perfect pairing (Poincare duality) on M induces an isomorphism of free rank 1 T` -modules β : M`-crys ∼ = HomO` (M`-crys , O` ), β(x)(y) = < x, y >, where the T` -module structure of HomO` (M`-crys , O` ) is induced by that of M`-crys . β is T` -linear as β(tx)(y) =< tx, y >=< x, ty >= (tβ(x))y, 46 where the central equality results from the fact that Hecke operators are selfadjoint with respect to the pairing. Define α : HomT` (M`-crys , T` ) → HomO` (M`-crys , O` ), by α(f ) = φ ◦ f . To see that α is an isomorphism, we assume that M`-crys = T` ⊕ T` without loss of generality. Then we have commutative diagram: HomT` (M`-crys , T` ) ∼ = ↓α HomT` (T` , T` ) ⊕ HomT` (T` , T` ) ↓ α1 ⊕ α1 HomO` (M`-crys , O` ) ∼ = HomO` (T` , O` ) ⊕ HomO` (T` , O` ), where α1 (f ) = φ ◦ f. Note that α1 = γ ◦ (f 7→ f (1)) is an isomorphism. Hence so is α. More over, α is T` -linear. Consider T` -isomorphism α−1 ◦ β. It corresponds to the T` -bilinear pairing << −, − >> on M`-crys defined by << x, y >>= ((α−1 ◦ β)(x))(y). Then φ(<< x, y >>) = (φ ◦ (α−1 ◦ β(x)))(y) = β(x)(y) =< x, y > . Let Fil1 be a shorthand for Fil1 M`-crys . Since M`-crys / Fil1 is T` free, we can pick a T` -submodule Comp of M`-crys so that M`-crys = Fil1 ⊕ Comp. We identify M`-crys / Fil1 with Comp. Note that Fil1 and Comp are free rank 1 T` -modules. 47 Proposition 3.3.2. << Fil1 , Fil1 >>= 0 Proof. For all x, y ∈ Fil1 , πτ << x, y >>= φ(eτ << x, y >>) = φ << eτ x, y >>=< eτ x, y >= 0. The last equality holds as Fil1 is a totally isotropic T` -submodule of M`-crys . Since τ is arbitrary, << x, y >>= 0. We have T` -isomorphism α−1 β : M`-crys ∼ = HomT` (M`-crys , T` ), or more explicitly, α−1 β : Fil1 ⊕ Comp ∼ = HomT` (M`-crys , T` )⊕HomT` (Comp, T` .) Once we choose a T` -base for each of the four free rank 1 T` -modules in the last congruence, the matrix of α−1 β is of the form    0 ∗  ,   ∗ ∗ where the upper left 0 is implied by the claim above. Since α−1 β is an isomorphism, the above matrix is invertible. In particular, the lower left element of the matrix must be a unit in T` . That is, if we let θ = α−1 β|Fil1 →HomT (Comp, T` ) , ` θ is a T` -isomorphism. 48 By slight abuse of notation, let the same symbol α denote the map HomT` (Comp, T` ) → HomO` (Comp, O` ) such that α(f ) = φ ◦ f. As before, α is a T` -isomorphism. Let v = α ◦ θ : Fil1 → HomO` (Comp, O` ), still a T` -isomorphism. Check that v(x)(y) = φ << x, y >>. On the other hand, fixing a T` -basis {b} for Comp thereafter, we can define v0 : Fil1 → HomO` (Comp, O` ) by v0 (f )(tb) = κ(f )(t). Since v0 and v are isomorphisms between free rank 1 T` -modules, we have v = v0 for some  ∈ T× ` . Since the desired result does not depend on the choice of ω, we will prove
the result by choosing a particular ω and showing that < f τ , ω(f τ ) > τ and µ∆∅ are actually equal for this choice. In fact, we choose ω : Fil1 → Comp such that ω(x) = ((γκ(x))−1 )b. It is seen that ω is a T` -isomorphism. So < f τ , ω(f τ ) > = φ << f τ , ω(f τ ) >> = v(f τ )(ω(f τ )) = ((v0 )f τ )(((γκ(f τ ))−1 )b) = κ(f τ )((γκ(f τ ))−1 ) = πτ (eτ ). 49
Hence to prove that < f τ , ω(f τ ) > τ and µ∆∅ are equal, is equivalent to showing that for all τ ∈ IT , πτ (eτ ) = πτ ( X eσ ). σ∈IT Now we prove the following proposition, thus ending the whole proof. Proposition 3.3.3. πτ (eσ ) = 0 for σ 6= τ . Proof. Since eσ = γ(κ(f σ )), where γ and κ are T` linear, the fact that f σ is in the πσ -eigenspace of Fil1 is translated to be that eσ is in the πσ -eigenspace of T` . Choose some element t ∈ T` such that πσ (t) 6= πτ (t). Then teσ = πσ (t)eτ . Apply πτ to both sides. As πτ is an algebra homomorphism, left side becomes πτ (t)πτ (eσ ). As πτ is O` -linear, the right side becomes πσ (t)πτ (eσ ). Now comparing both sides leads to the desired claim. 50 Chapter 4 A non-trivial example Remark 1.4.3 says that the exact extent to which the conjecture of Burns-Flach × is finer than that of Bloch-Kato in our case is measured by the group O× T` /T` . To show that what we have proved is not vacuous, it is imperative that we have a concrete example for which our proof of the conjecture of Burns-Flach works and that this group is nontrivial. We recall or introduce a few symbols first. T is the integral Hecke algebra generated over Z by the Hecke operators on S2 (Γ0 (N )), the vector space of cusp forms of weight 2, level prime number N and trivial character. Since N is prime, S2 (Γ0 (N )) is generated by newforms as there is no forms in S2 (Γ0 (1)) = S2 (SL2 (N )). T = T ⊗Z Q = Q f Kf , where f runs through non-conjugate newforms in S2 (Γ0 (Z)) and where Kf is the field of definition of f . SI is the set of prime number ` such that either: • λ | 2N , or • there exist a newform f and a prime λ | ` in Kf such that the twodimensional residual Galois representation Mf,λ /λMf,λ is not absolutely 51 irreducible when restricted to GF , where Mf is the premotivic structure p associated to f defined in [D-F-G2, §5.4] and F = Q( (−1)(`−1)/2 `) and λ | `. We want to choose N and ` to satisfy • condition 1: ` 6∈ SI , and • condition 2: (T` )× is a proper subgroup of O× T` . We set the first condition as we use the result of [D-F-G2, §8.2] to prove BurnsFlach conjecture in our case while the second condition is the requirement of non-trivial refinement. Example: The above two conditions are satisfied for N = 89 and ` = 5. S2 (Γ0 (89)) contains 7 (normalized) newforms, 5 of which are Galois conjugate to each other. Let f1 , f2 , f3 be three non-Galois-conjugate (normalized) newforms in S2 (Γ0 (89)). We list in table 4.1 the eigenvalues of the first 15 Hecke operators acting on them, or equivalently, the first 15 coefficients of their q-expansions, which can be found by [St3] or in [St1]. Now let us check condition 1. Suppose it is not satisfied. Since 5 - 2 · 89, there is a prime λ | 5 in Sf where f = fi for some i ∈ 1, 2, 3 such that the two-dimensional residual Galois representation Mf,λ /λMf,λ is not absolutely irreducible when restricted to GF . By [D-F-G2, Lemma 7.14], the original representation is not absolutely irreducible either. By the proof of [D-F-G2, 52 Lemma 7.13] , we must have ap (f ) ≡ p + 1 mod λ (4.1) for all primes p ≡ 1 mod N. Choose p = 2 ∗ 89 + 1 = 179. Then p + 1 ≡ 0 mod λ. The following values of a179 ’s can be found the same way as above. a179 (f1 ) = 14 6≡ 0 mod λ a179 (f2 ) = 4 6≡ 0 mod λ a179 (f3 ) = (−4a4 + 4a3 + 32a2 − 16a − 42) 6≡ 0 mod λ. The last inequality comes from the fact that the norm of a179 (f3 ) is 413408, which is not divisible by 5 and which is obtained by software package Pari as follows: ?norm(Mod(-4*a^4+4*a^3+32*a^2-16*a-42,a^5+a^4-10*a^3-10*a^2+21*a+17)) \%1=413408 Hence, (4.1) is not true for any f . So condition 1 is satisfied. Now let us check condition 2. By [A-S], the integral Hecke algebra T is generated (as an abelian group) inside Q × Q × Q(a) by the operators Tn with n ≤ 2 · 89/12 · (1 + 1/89) = 15, i.e., generated by the rows of table 4.1 (of course not considering the first row and first column). By row operations, we 53 n an (f1 ) an (f2 ) an (f3 ) 1 1 1 1 2 -1 1 a 4 3 3 -1 2 −a /2 + a /2 + 7a2 /2 − 5a/2 − 4 4 -1 -1 a2 − 2 5 -1 -2 −a2 + 4 6 1 2 a4 − 3a3 /2 − 15a2 /2 + 13a/2 + 17/2 7 -4 2 a4 /2 − 4a2 − a + 13/2 8 3 -3 a3 − 4a 9 -2 1 a2 − a − 4 10 1 -2 −a3 + 4a 11 -2 -4 −a3 + 5a + 2 12 1 -2 −3a4 /2 + 3a3 /2 + 19a2 /2 − 15/2a − 9 13 2 2 −a4 + a3 + 8a2 − 5a − 11 14 4 2 −a4 /2 + a3 + 4a2 − 4a − 17/2 15 1 -4 a4 /2 − a3 /2 − 5a2 /2 + 5a/2 + 1 Table 4.1: Hecke operator Tn on newforms in S2 (Γ0 (89)) Here a is a root of a5 + a4 − 10a3 − 10a2 + 21a + 17 = 0. find that T is also generated by 7 linearly-independent elements (1, 0, 5) (0, 1, −4) (0, 0, (a4 + 1)/2 − 1) (0, 0, (a3 + a2 + a + 1)/2 + 3) (0, 0, a2 − 1) (0, 0, a − 1) (0, 0, 10) in OT ∼ = Z × Z × OKf3 . Note that {(a4 + 1)/2, (a3 + a2 + a + 1)/2, a2 , a, 1} is a Z-basis of OKf3 as shown by the following Pari session: 54 ? nfinit(a^5+a^4-10*a^3-10*a^2+21*a+17).zk %1 = [1, a, a^2, 1/2*a^3 + 1/2*a^2 + 1/2*a + 1/2, 1/2*a^4 + 1/2] The set of 7 elements above also generates T5 = T ⊗Z Z5 as a Z5 -module. Since the unit (1, 1, −1) ∈ OT5 is not in T5 , we have verified condition 2. × For completeness, let us determine the quotient O× T` /T` in this case. As 2 is invertible in Z5 , OKf3 ,5 as a Z5 -module is generated by {a4 − 1, a3 − 1, a2 − 1, a − 1, 1}, or equivalently, by {b4 , b3 , b2 , b, 1} with b = a − 1. The ideal (5, b) in OKf3 ,5 is generated by {b4 , b3 , b2 , b, 5} as a Z5 -module as b5 = −6b4 − 4b3 + 24b2 + 20b − 20. Hence the map α : Z5 /(5) → OKf3 ,5 /(5, b) induced by the inclusion Z5 → OKf3 ,5 is a ring isomorphism. T5 is generated by (1, 0, 0), (0, 1, 1), (0, 0, 5), (0, 0, b), (0, 0, b2 ), (0, 0, b3 ), and (0, 0, b4 ), while OT5 is generated by the same set except that (0, 0, 5) is replaced by (0, 0, 1). So, T5 is exactly the elements (x1 , x2 , x3 ) in OT5 such that × × α(x2 ) = x3 . As OT5 is an integral extension of T5 , T× 5 = OT5 ∩T5 . Hence, T5 is exactly the elements (x1 , x2 , x3 ) in O× T5 such that α(x2 ) = x3 , or, equivalently, the kernel of the homomorphism × × × × O× T5 = Z5 × Z5 × OKf ,5 → (Z5 /5Z5 ) 3 (x1 , x2 , x3 ) 7→ x2 α−1 (x3 ). × ∼ × Since this homomorphism is surjective, O× T5 /T5 = (Z5 /5Z5 ) . Its order, 4, indicates that all we have done is a small refinement indeed. We wonder if 55 there is an easier way to achieve it. 56 Chapter 5 An isomorphism of two Selmer groups One type of Selmer groups is defined by Bloch and Kato and employed in their conjecture while another type is given in [D-F-G2, §7.1] and reintroduced in 2.1. This chapter presents in proposition 5.3.4 an isomorphism between these two types under some conditions, which can be viewed as a kind of global version of the isomorphism in [D-F-G2, §7.1], whose statements and proofs are followed quite closely here. The comparison of Selmer groups is considered one of the first steps to understand these groups. In the course of proving the isomorphism, we also obtain proposition 5.1.1 on the A-linear FontaineLaffaille theory, where A is a commutative complete local noetherian ring finite over Z` . This chapter is independent of the other chapters except that proposition 5.1.1 is mentioned in §3.1. 57 5.1 The theory of Fontaine and Laffaille We introduce some of the theory of Fontaine and Laffaille in [F-L] in a way that fits for later use. The main point here is to extend the coefficients of various modules to A or A/`, where A is a commutative complete local noetherian ring finite over Z` . By common abuse of notation, we use the same symbol to denote an object of a category or its image under a forgetful functor when it is clear which one we refer to. Let Z` -MF, or simply MF, denote the category whose object X is a finitely generated Z` -module equipped with • a decreasing filtration such that Fila X = X and Filb X = 0 for some a, b ∈ Z, and for each i ∈ Z, Fili X is a Z` -module direct summand of X; • Z` -linear maps φi : Fili X → X for i ∈ Z satisfying φi |Fili+1 X = `φi+1 and X= P Imφi , and whose morphisms are Z` -module homomorphisms respecting filtration and commuting with φi for all i. MF is naturally a Z` -linear category. It follows from [F-L, 1.8] that MF is an abelian category such that the forgetful functor from it to the category of Z` -modules is exact. For any subcategory C of MF, let C0 denote its full subcategory of objects X satisfying Fil0 X = X and Fil` X = 0 and having no non-trivial quotients X 0 such that Fil`−1 X 0 = X 0 . Also, let MFtor denote its full subcategory of 58 objects of finite length, i.e., objects that are torsion Z` -modules. It follows from [F-L, 6.1] that MF0tor is an abelian category, stable under taking subobjects, quotients, finite direct products and extensions in MF. Now we extends the coefficients to A or A` . Let A-MF denote the category any of whose object is an object X in MF equipped with a ring homomorphism A → EndMF (X), a 7→ aX for all a ∈ A, which becomes the structure map Z` → EndMF (X) if composed with the natural map Z` → A, and whose morphisms should be compatible with the extra structure, namely, for X and X 0 in A-MF, a morphism f : X → X 0 in Z` -MF is also a morphism in A-MF if and only if f · aX = aX 0 · f for all a ∈ A. Explicitly, an object of A-MF is a finitely generated A-module X equipped with • a decreasing A-module filtration such that Fila X = X and Filb X = 0 for some a, b ∈ Z, and for each i ∈ Z, Fili X is a Z` -module direct summand of X; • A-linear maps φi = φiX : Fili X → X for i ∈ Z satisfying φi |Fili+1 X = `φi+1 and X = P Imφi , and a morphism is an A-module homomorphism respecting filtration and commuting with φi for all i. Substituting A/` for A in the previous paragraph, we obtain the category A/`-MF. A-MF0 and A/`-MF0 are the full subcategories of A-MF and A/`-MF respectively whose objects are actually in MF0 . We note that the description of an object in A-MF is the same as the 59 description of an object in Z` -MF with Z` replaced by A everywhere with one exception. The following proposition reveals that under a mild condition, there is no exception. The proposition still holds with almost the same proof even if A is not commutative. However, this is irrelevant to this article. Proposition 5.1.1. Let X be an object of A-MF whose underlying module is a free A-module. Then Fili X is in fact an A-module direct summand of (the underlying module of ) X and a free A-module for all i. Proof. We do induction on the nontrivial filtration length n of X. If n = 0, i.e., X = 0, it is trivial. Suppose it is true if the nontrivial filtration length is less than n. Let X ∈ A-MF whose non-trivial filtration length is n. Shifting the filtration indices if necessary, we can assume X = Fil0 X ⊃ Fil1 X · · · ⊃ Filn = 0. We first deal with Fil1 X. Let M be the matrix representation of φ0 with respect to a basis of X over A. Since A is a local ring, using elementary row and column operations, we know that there exist two invertible matrices Mα and Mβ such that   I 0  , Mα M Mβ =    0 M1 where I is the identity matrix and M1 is a square matrix whose entries are in M, the maximal ideal of A. Let Mα and Mβ correspond to A-linear isomorphisms α and β respectively. We identify the domain of β with the codomain of α in such a way that the bases for them coincide. Replacing Fili X by 60 β −1 (Fili X) and φi by α ◦ φi ◦ β for all i, we can assume from now on that M has the block diagonal form of the right hand side of above equality. Let X = X0 ⊕X1 such that it corresponds to the block form of M . Consider x ∈ N = X0 ∩ Fil1 X. Then x = φ0 (x) = `φ1 (x). Since X0 and Fil1 X are Z` module direct summands of X, φ1 (x) ∈ N . So x ∈ `N, i.e., N ⊂ `N. Note that ` ∈ M. By Nakayama’s lemma, N = 0. Hence, rankZ` X0 + rankZ` Fil1 X = rankZ` (X0 + Fil1 X) ≤ rankZ` X. We have X = (5.1) i i≥0 Imφ . Then P P Imφ0 X0 + i≥1 Imφi X1 = + . X0 X0 Note that Imφ0 /X0 = φ0 (X1 ) ⊂ MX1 . By Nakayama’s lemma, X = X0 + X i i≥1 Imφ . (5.2) So we have the first inequality of the following: rankZ` X ≤ rankZ` X0 + rankZ` i i≥1 Imφ P = rankZ` X0 + rankZ` Imφ1 (5.3) ≤ rankZ` X0 + rankZ` Fil1 X, where the second equality is ensured by the fact that φi |Fili+1 X = `φi+1 for all 61 i. Combining (5.1) with (5.3), we see that all inequalities must be equalities and hence, rankZ` X = rankZ` X0 + rankZ` Fil1 X, (5.4) which leads to equality: dimF` X X0 Fil1 X = dimF` + dimF` , `X `X0 ` Fil1 X (5.5) where we can naturally identify each quotient on the right hand side with an F` -vector subspace of the quotient on the left hand side as X0 and Fil1 X are Z` -module direct summands of X. The intersection of those two quotients on the right is zero. To prove this claim, let us take an arbitrary element in the intersection, which is x0 + `X for some x0 ∈ X0 and x1 + `X for some x1 ∈ Fil1 X. So x0 = x1 mod `X. Applying the natural projection π : X → X0 to this congruence equality, we have x0 = π(x1 ) mod `X0 , because π(`X) ∩ X0 = `X0 . On the other hand, π(x1 ) = φ0 (π(x1 )) = πφ0 (x1 ) = π`φ1 (x1 ) ∈ π(X) ∩ `X = `X0 . So x0 = 0 mod `X0 and the claim is established. Then from (5.5), we see 62 that X X0 Fil1 X + . = `X `X0 ` Fil1 X Nakayama’s lemma tells us that X = X0 + Fil1 X. Then by (5.4), we must have X = X0 ⊕ Fil1 X. So Fil1 X is an A-module direct summand of X. Since a projective module over a local ring is free, Fil1 X is a free A-module. So the desired conclusion is true for Fil1 X. (5.2) and the first inequality in (5.3), where all inequalities are actually equalities as shown before, imply that X = X0 ⊕ projective module over a local ring, i i≥1 Imφ . Thus being a P i i≥1 Imφ is a free A-module. It has the P same rank over A as Fil1 X as shown by the last (in)equality in (5.3). Let γ be an A-linear isomorphism from 1 i i≥1 Imφ to Fil X. P Construct an object Y of A-MF from X as follows. Y is Fil1 X equipped with extra structure: • Fili Y = Fil1 X for all i ≤ 1, Fili Y = Fili X for all i ≥ 1, • φi−1 = `φiY for all i ≤ 1, φiY = γ ◦ φiX for all i ≥ 1. Y Then Y is an object of A-MF whose nontrivial filtration length is less than n. By induction assumption, Fili Y is an A-module direct summand of Fil1 Y and a free A-module for all i. In other words, Fili X is an A-module direct summand of Fil1 X and a free A-module for all i ≥ 1. Since Fil1 X is an A-module direct summand of X, Fili X is an A-module direct summand of X. So the desired conclusion for Fili X is true if i ≥ 1, while it is trivially true if i ≤ 0. Thus we 63 conclude our induction. Let Z` -GR, or simply GR, be the full subcategory of Z` G` -modules whose objects are isomorphic to quotients of the form L1 /L2 , where L2 ⊂ L1 are finitely generated G` -stable Z` -submodules in short crystalline representations. In the same way as we define A-MF and A/`-MF from MF, we define category A-GR and category A/`-GR from GR. The category MF0tor is Z` -linearly equivalent, via the functor, also mentioned at §1.2, X 7→ V(X) := hom(U S (X), Q` /Z` ), to FR, where U S (X) = hom(some appropriate category) (X, Acrys,` ⊗Z` Q` /Z` ), and FR, already introduced in §2.1, is the full subcategory of GR of objects of finite length [F-L, 6.1]. Extend V to a fully faithful functor on MF0 by setting V(X) = proj lim V(X/`n X). Then V defines an equivalence between MF0 and GR. By functoriality, the functor V also defines an A-linear equivalence between A/`-MF0 and A/`-GR and between A-MF0 and A-GR. We know GR and FR are abelian categories. Note that we can describe A-GR (resp. A/`-GR) as the full subcategory of the abelian category of finitely–generated–over–Z` AG` -modules, consisting of objects that fall into GR (resp. FR) under the exact forgetful functor to the abelian category of finitely–generated–over–Z` Z` G` -modules. (The condition of being finite Z` modules is to guarantee that the categories are abelian so that the argument 64 right below would work. Note that the modules here are, by the convention throughout this article, modules with continuous G` action on its Z` -adic topology.) Therefore A-GR and A/`-GR are abelian categories. By equivalence of categories, so are A/`-MF0 and A-MF0 . Note that the forgetful functor from A/`-GR to GR is exact. By equivalence of categories, the forgetful functor from A/`-MF0 to MF0 is exact. Hence, the forgetful functor from A/`-MF0 to the category of A-modules is exact. The same arguments exist for A-MF0 . In a word, A-MF0 , A/`-MF0 , A-GR and A/`-GR are A-linear abelian categories such that the forgetful functors from them to the category of A-modules are exact. Now we state a proposition that applies to V. We know that V, as a functor on MF0 , preserves the lengths of underlying Z` -modules. For any A-module, its length as a Z` -module is [k : F` ] times its length as an A-module, where k is the residue field of A. So V, as a functor on A-MF0 , also preserves the lengths of underlying A-modules. Proposition 5.1.2. Let A be a commutative complete local noetherian ring. Suppose we are given two A-linear abelian categories such that there exist exact forgetful functors from them to the category of A-modules, where the A-module structure of an object is provided by its original A-linear structure. Then an equivalence between them that preserves lengths of the underlying A-modules also preserves the freeness and ranks of the underlying A-modules (of objects 65 whose underlying A-modules are free). Proof. Let Θ : E → F be the equivalence in the proposition and M be the maximal ideal of A. Suppose Θ(E) = F , where F is a free A-module of rank r. For any ideal I of A generated by a1 , · · · , an , we have an exact sequence in the category of A-modules: β E n → E → E/IE → 0, where β((x1 , · · · , xn )) = P ai xi , xi ∈ E. β is, in fact, in E because E is an A-linear abelian category. The exact forgetful functors of E to the category of A-modules ensures that E/IE, the cokernel of β in the category of A-modules, must be (the underlying A-module of) the cokernel of β in E. So we can view the above sequence as exact in E. Applying Θ, we get an exact sequence in F: Θ(β) F n → F → Θ(E/IE) → 0, Θ(β) where Θ(β) has the same definition as β. Since the cokernel of F n → F in F is F/IF for the same reason as above, we obtain that Θ(E/IE) ∼ = F/IF. Hence as A-modules, length(E/IE) = length(Θ(E/IE)) = length(F/IF ) = r · length(A/I), (5.6) 66 where lengths might be infinity. Let M take the place of I. Then (5.6) becomes dimA/M (E/ME) = dimA/M (F/MF ) = r. By Nakayama’s lemma, we have a surjection of A-modules: Ar → E, which is completed to an exact sequence: 0 → K → Ar → E → 0. Tensoring it with A/Mk over A for an integer k, we get an exact sequence K ⊗A A/Mk → (A/Mk )r → E/Mk E → 0. So we have length( image of K ⊗A A/Mk ) = r · length(A/Mk ) − length(E/Mk E). One can show that length(A/Mk ) is finite by induction as A is noetherian. Hence, (5.6) says that the right hand side of the above equality is 0. Then K ∈ Mk · Ar . Let k goes to infinity, we see that K = 0 as A is complete with respect to M. So Ar ∼ = E. Remark 5.1.3. All propositions still holds if we change A to be a finite product of commutative complete noetherian local rings finite free over Z` , or, what is 67 exactly the same, a commutative semi-local ring finite free over Z` . This can be seen by decomposing into or taking product over components indexed by the Wedderburn factors of A for various concepts. 5.2 Notation We continue to use the notation of §2.1. Fix an odd prime `, a commutative complete local ring A finite free over Z` , a continuous AGQ -module L free of rank two over A and W = End0A (L) ⊗Z` Q` /Z` , where the superscript 0 means endomorphisms of trace 0. Let T =End0A (L), B = A ⊗Z` Q` , V1 = L ⊗Z` Q` , and V = T ⊗Z` Q` = End0B (V1 ). So we have an exact sequence 0 → T → V → W → 0. (5.7) We assume that V1 is a short crystalline representation of G` , where the meaning of short is defined in §2.1. Bloch and Kato define a divisible B-submodule Hf1 (Gp , W ) ⊂ H 1 (Gp , W ) for each p. Explicitly, Hf1 (Gp , V ) :=     ker(H 1 (Gp , V ) → H 1 (Ip , V )) = H 1 (GFp , V Ip ) for p 6= `,    for p = `, ker(H 1 (G` , V ) → H 1 (G` , Bcrys ⊗Q` V )) where Bcrys is the ring Bcrys,` defined by Fontaine [F-P2, I.2.1]. Then Hf1 (Gp , W ) 68 is the image of Hf1 (Gp , V ) under the natural map. Their Selmer group Hf1 (GQ , W ) ⊂ H 1 (GQ , W ) is the A-submodule of elements with restrictions in Hf1 (Gp , W ) for all primes p. 5.3 The isomorphism of two Selmer groups We first prepare two isomorphisms in the next two propositions. Proposition 5.3.1. Let p 6= ` and suppose W Ip is divisible. Then Hf1 (Gp , W ) = Hf1 (Gp , W ), where Hf1 (Gp , W ) is defined in §2.1. Proof. We have a long exact sequence: δ 0 → T Ip → V Ip → W Ip → H 1 (Ip , T ) → · · · . Im(δ) is divisible as so is W Ip . But H 1 (Ip , T ) has no nontrivial `-divisible subgroups by [N-S-W, Proposition 2.3.7]. So Im(δ) = 0. Replacing H 1 (Ip , T ) by 0 in the above sequence, we have a short exact sequence of Gp /Ip = GFp - 69 modules, which induces a long exact sequence: β α · · · → H 1 (GFp , V Ip ) → H 1 (GFp , W Ip ) → H 2 (GFp , T ) → · · · What we want to prove is none other than that α is surjective. So it suffices to show H 2 (GFp , T ) = 0. We have H 2 (GFp , T ) = lim H 2 (GFp , T /`n T ) = 0. ←− The first equality holds by [N-S-W, Corollary 2.3.5] and the second by Proposition 1.6.13(ii) of loc. cit. Proposition 5.3.2. Hf1 (G` , W ) = Hf1 (G` , W ) is divisible of Z` -corank d = dimQ` H 0 (G` , V ) + dimQ` V − dimQ` Fil0 Dcrys (V ). Here Dcrys (V )= (Bcrys ⊗Q` V )G` . The Z` -corank of a Z` -module M is the rank of M ∨ /(M ∨ )tor over Z` , where M ∨ = homZ` (M, Q` /Z` ). Proof. First we show Hf1 (G` , W ) ⊂ Hf1 (G` , W ). Let α ∈ Hf1 (G` , W ) be the image of some class c̄ ∈ Hf1 (G` , V ) represented by cocycle c ∈ Z 1 (G` , V ), which corresponds to a BG` -module extension 0 → V → E → B → 0, (5.8) 70 where B has trivial G` action and E = V ⊕ B with G` action g(v, b) = (g(v) + bc(g), b). We construct a commutative diagram of exact rows 0 → V ⊗B V1 0 → → ↓ push-out ↓ V1 → E0 ↑ 0 → L L/`n L ↓ → ↑ → ↓ 0 → E ⊗B V1 → B ⊗B V1 → 0 L0 L0 /`n L0 → 0 ↑ → ↓ → V1 L → 0 ↓ → L/`n L → 0 as follows. The first row, (5.8)⊗B V1 , is exact as V1 is free over B. The upper left vertical arrow is the evaluation map. Then we make the upper left square a BG` -module push-out diagram and complete first two rows so that it commutes and that the second row is exact. See [Ei, A3.26c]. We require that the third row be exact and the arrows from it to the second row are natural injections. Then L0 is a uniquely determined free A-module. The arrows to the fourth row are natural projections. Then the fourth row is also exact. We pick n large enough so that Im(c) ⊂ T ⊗Z` `−n Z` /Z` . Then the map from c̄ to α factors through the image of c in H 1 (G` , Wn ). Call this image β. (By the way, if we know E 0 , we can get E by constructing the following 71 commutative diagram with exact rows: 0 → V ⊕B → F → ↓ ↓ (pull-back) ↓ → V ⊕B 0 → V ⊕ B → homB (V1 , E 0 ) B · idV1 → 0 → 0, where V ⊕ B = EndB (V1 ). Then E = F/B.) We have defined Hf1 (G` , Wn ) using the one-one correspondence between elements of H 1 (G` , Wn ) and Yoneda extensions of L/`n L by itself in the category of Z` /`n G` -modules. A long but routine check demonstrates that the fourth row and β corresponds to each other under that correspondence. Hence, to show α ∈ Hf1 (G` , W ), it is enough to show β ∈ Hf1 (G` , Wn ), or equivalently by definition, that E 0 is a short crystalline representation since L0 is a G` -stable Q` -lattice in E 0 . By the following lemma, we see that E 0 is crystalline if E is crystalline, a fact which we will prove right after the proof of the lemma. Since V1 is short, so is E 0 . Lemma 5.3.3. Crystalline representations (over Q) are stable under taking finite direct sum, subobject, quotient object, internal homomorphism and tensor product over B in the category of (finitely generated) BG` -modules. Proof. We know that crystalline representations are stable under taking finite direct sum, subobject, quotient object, internal homomorphism and tensor product in the category of Q` G` -modules. So they are also stable under taking finite direct sum, subobject, quotient object in the category of BG` -modules 72 as the forgetful functor from this category to the category of Q` G` -modules is exact. The internal homomorphisms of two objects in the category of BG` modules is a subobject of the internal homomorphisms of these two object in the category of Q` G` -modules. The tensor product of two objects in the category of BG` -modules is a quotient object of the tensor product of these two objects in the category of Q` G` -modules. Hence, crystalline representations are also stable under taking internal homomorphism and tensor product in the category of BG` -modules. Now the lemma is proved. So V is also crystalline. We have a commutative diagram with exact rows: 0 → → V → E ↓ ↓ → 0 B ↓ → Bcrys ⊗Qp E → Bcrys ⊗Qp B → 0, 0 → Bcrys ⊗Qp V which induces a commutative diagram of cohomology groups with exact rows: 0 → V G` ↓ → E G` ↓ → B G` r → ↓ H 1 (G` , V ) → ↓s t 0 → Dcrys (V ) → Dcrys (E) → Dcrys (B) → H 1 (G` , Bcrys ⊗Q` V ) →, 73 where the third vertical arrow is just the identity map of B. So dimQ` (Dcrys (E)) = dimQ` (Dcrys (V )) + dimQ` (Dcrys (B)) − dimQ` (Im(t)) = dimQ` V + dimQ` B − dimQ` (< t(1 ⊗Q` 1) >B−module ) = dimQ` E − dimQ` (< s(r(1)) >B−module ) Note that r(1) = c̄. By definition of Hf1 (G` , V ), s(c̄) = 0. So E is crystalline. The divisibility of Hf1 (G` , W ) follows from its definition, and its Z` -corank is equal to the dimQ` Hf1 (G` , V ), which is computed to be d in [B-K, §3.8.4] or [F-P2, §3.3 of Chapter 3]. We explain why the corank is equal to the dimQ` Hf1 (G` , V ). We have a commutative diagram H 1 (G` , T ) ⊗Z` Q` ∼ = H 1 (G` , V ) ↓ ↓ 0 → H 1 (G` , T ) ⊗Z` Q` /Z` → H 1 (G` , W ) ↑ ↑ 0 → H 1 (G` , T )/`m → H 2 (G` , T )tor → 0 ↑ → H 1 (G` , T /`m ) → H 2 (G` , T )[`m ] → 0, where the third row is exact and the second row, as the direct limit of the third row, is also exact. The group H 2 (G` , T )tor is finite. See [N-S-W, §2.3.7to §2.3.10] for all non-trivial facts in the diagram. The first two rows show H 1 (G` , W ) is nearly a co-lattice in H 1 (G` , V ) in a sense that can be made clear and hence so is Hf1 (G` , W ) in Hf1 (G` , V ). 74 To finish the whole proof, it suffices to prove that dimF` Hf1 (G` , W )[`] ≤ d. Recall that finite AG` -module W1 is defined by the exact sequence ` 0 → W1 → W → W → 0, which induces the exact sequence 0 → F` ⊗Z W G` → H 1 (G` , W1 ) → H 1 (G` , W )[`] → 0. Since Hf1 (G` , W1 ) is the preimage of Hf1 (G` , W ) of map H 1 (G` , W1 ) → H 1 (G` , W ), we derive another exact sequence 0 → F` ⊗Z W G` → Hf1 (G` , W1 ) → Hf1 (G` , W )[`] → 0. So we have the first equality of the following: dimF` Hf1 (G` , W )[`] = dimF` Hf1 (G` , W1 ) − dimF` (F` ⊗Z` W G` ) = dimF` Hf1 (G` , W1 ) − dimF` W1G` + dimQ` H 0 (G` , V ). The second equality holds as once we write W G` ∼ = (Q` /Z` )m ⊕ U with U a finite torsion Z` -module, then dimF` (F` ⊗Z` W G` ) = dimF` U/`U = dimF` U [`], dimF` W1G` = dimF` (`−1 Z` /Z` )m ⊕U [`] = m+dimF` U [`] and dimQ` H 0 (G` , V ) = 75 m. The last equality can be proved as in the following paragraph. As H 0 (G` , V ) = H 0 (G` , T ) ⊗Z` Q` , dimQ` H 0 (G` , V ) = dimZ` H 0 (G` , T ). It is clear that dimZ` H 0 (G` , T ) ≤ m. Suppose we are given wi ∈ W G` [`i ] corresponding to (1/`i , 0, · · · , 0; 0). Then wi = wi0 ⊗ 1/`i for wi ∈ T. If j > i, pj−i wj = wi . Hence, (wj0 − wi0 ) ⊗ 1/`i = 0, i.e., wj0 − wi0 ∈ `i T . Hence limi wi0 converges to some element w in T . w is in T G` as action by G` is continuous on T. Continuing in this way, we can construct m elements in T G` that are Z` -linear independent. So dimZ` H 0 (G` , T ) ≥ m. Hence the desired equality. Therefore it suffices to prove that dimF` Hf1 (G` , W1 ) − dimF` H 0 (G` , W1 ) = dimQ` V − dimQ` Fil0 Dcrys (V ). (5.9) Choosing an object D in A-MF0 such that V(D) ∼ = L ∈ A-GR. Proposition 5.1.2 shows that D is a free A-module. Proposition 5.1.1 tells further that Fili D is an A-module direct summand of D and a free A-module for all i. Let D/` be the cokernel of multiplication by ` on D. By the equivalence of the categories, V(D/` ) ∼ = L/`L. We have Fili D/` = Fili D/(Fili D ∩ `D) = Fili D/` Fili D, the latter equality ensured by the fact that Fili D is a Z` -module direct summand of D. We now construct the following exact sequence 0 → homA/`-MF (D/` , D/` ) → homA/`,Fil (D/` , D/` ) 1−φ → homA/` (D/` , D/` ) π → Ext1A/`-MF (D/` , D/` ) (5.10) → 0, 76 where Ext1 is the abelian group of Yoneda extensions in an abelian category with enough objects, like push out, pull back, etc. The first nonzero map is a part of the forgetful functor. Since each filtration of D is an A-module direct summand of D, each filtration of D/` is also an A/`module direct summand of D/`. Hence we have Fili D/` = Di/i+1 ⊕ Fili+1 D/` for some choice of an A/`-module Di/i+1 . Note that D/` = ⊕Di/i+1 and that φi (Fili+1 D/` ) = 0. Define φ ∈ EndA/` (D/` ) such that φ|Di/i+1 = φi . Then φ(D/` ) = ΣIm(φi ) = D/` . So φ is an isomorphism as D/` has finite length as Z` -modules. By abuse of notation, let φ also denote its adjoint action on EndA/` (D/` ) so that the second nonzero map is defined. Now we explain the construction of π. For η ∈ homA/` (D/` , D/` ) we define an extension Eη of D/` by itself in A/`-MF with underlying A/`-module D/` ⊕ D/` , filtration Fili Eη := Fili D/` ⊕ Fili D/` and Frobenius map φi : Fili Eη → Eη φi (x, y) = (φi (x) + ηφi (y), φi (y)). Then π(η) is the class of the Yoneda extension Eη in Ext1 . The verification of the exactness of the sequence (5.10) is straightforward. (To show π is a surjection, we use the fact that D/` is free over A/`, which is true as D is free over A.) 77 Here comes a small twist of proof. We can redo the proof from the very beginning up to now with trivial modifications if we consider EndA (L) instead f = Te ⊗ /Z` , and Ve = Te ⊗Z Q` , etc of End0A (L), i.e., consider Te = EndA (L), W ` while `, A, L, D are the same as before. Now we verify (5.10) implies (5.9). f1 ) = {Yoneda extensions of L/` by itself in A/`-GR} Hf1 (G` , W = {Yoneda extensions of D/` by itself in A/`-MF0 } = Ext1A/`-MF (D/` , D/` ) f1 ) = (homA/` (L/`L, L/`L))G` H 0 (G` , W = homA/`G` (L/`L, L/`L) = homFR(A/`) (L/`L, L/`L) = homA/`-MF (D/` , D/` ) dimQ` Ve = dimQ` homB (V1 , V1 ) = dimZ` homA (L, L) = dimZ` homA (D, D) = dimF` homA/` (D/` , D/` ) The second last equality holds because L and D are free over A of the same 78 rank by Proposition 5.1.2. dimQ` Fil0 Dcrys (Ve ) = dimQ` Fil0 (Bcrys ⊗Q` (Q` ⊗Z` EndA (L))G` = dimQ` Fil0 (Q` ⊗Z` EndA (D)) = dimZ` Fil0 (EndA (D)) = dimZ` homA,Fil (D, D) = dimF` homA/`,Fil (D/` , D/` ) The last equality holds as Fili D/` are free over A/` of the same rank as Fili D over A. The fourth equality from the bottom requires more explanation. One sees from [F-L, 8.4] that for X ∈ MF, the natural map of filtered Galois φ-modules Bcrys ⊗Z` X → Bcrys ⊗Z` V(X) is an isomorphism, which leads to isomorphism of filtered φ-modules: Q` ⊗Z` X → (Bcrys ⊗Z` V(X))G` . (5.11) So we need to show that V(EndA L) = EndA D, which can be proved by tracing the definition of V. Note that EndA L is indeed an object of A-MF0 . f ) = H 1 (G` , W f ) holds. If we So we have proved the equality Hf1 (G` , W f f back to W , the equality still holds as W f = W ⊕ (A ⊗Z Q` /Z` ), change W ` Hf1 (G` , W ) ⊂ Hf1 (G` , W ), Hf1 (G` , A⊗Z` Q` /Z` ) ⊂ Hf1 (G` , A⊗Z` Q` /Z` ), Hf1 (G` , 79 f ) = H 1 (G` , W ) ⊕ H 1 (G` , A ⊗Z Q` /Z` ), and H 1 (G` , W f ) = H 1 (G` , W ) ⊕ W f f f f ` Hf1 (G` , A ⊗Z` Q` /Z` ), where A is identified with the maps from L to itself that are multiplication by elements of A. Combining Propositions 5.3.1 and 5.3.2, we arrive at the following desired equality . Proposition 5.3.4. 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